Finite Semifields and Projective Planes Citation Knuth, Donald Ervin (1963) Finite Semifields and Projective Planes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/T3Q6-JC64. https://resolver.caltech.edu/CaltechETD:etd-06042004-141331 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This paper makes contributions to the structure theory of finite semifields, i.e., of finite nonassociative division algebras with unit. It is shown that a semifield may be conveniently represented as a 3-dimensional array of numbers, and that matrix multiplications applied to each of the three dimensions correspond to the concept of isotopy. The six permutations of three coordinates yield a new way to obtain projective planes from a given plane. Several new classes of semifields are constructed; in particular one class, called the binary semifields, provides an affirmative answer to the conjecture that there exist non-Desarguesian projective planes of all orders 2[...], if n is greater than 3. With the advent of binary semifields, the gap between necessary and sufficient conditions on the possible orders of semifields has disappeared. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Turing Award, 1974. National Medal of Science, 1979. Harvey Prize, 1995. Kyoto Prize in Advanced Technology, 1996. Thesis Availability: Public (worldwide access) Research Advisor(s): Hall, Marshall Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Record Number: CaltechETD:etd-06042004-141331 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-06042004-141331 DOI: 10.7907/T3Q6-JC64 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2441 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 07 Jun 2004 Last Modified: 04 Jan 2024 18:58 Thesis Files Preview PDF (Knuth_de_1963.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

FINITE SEMIFIELDS AND PROJECTIVE PLANES Thesis by Donald Ervin Knuth In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 ACKNOWLEDGMENTS I am indebted to Professor Marshall Hall, Jr., for his inspiration and guidance as I pursued this work. Thanks are also due to California Institute of Technology for providing scholarships to support my studies. Many of the results of this paper were suggested and/or confirmed by calculations on a Burroughs 220 digital computer, and I am grateful to the Burroughs Corporation for providing computer time. ABSTRACT This paper makes contributions to the structure theory of finite semi fields, i.e., of Finite nonassociative division algebras with unit. It is shown that a semifield may be conveniently represented as a 3~dimensional array of numbers, and that matrix multiplications applied to each of the three dimensions correspond to the concept of isotopy. The six permutations of three coordinates yield a new way to obtain projective planes from a given plane. Several new classes of semifields are constructed; in particular one class, called the binary semifields, provides an affirmative answer to the conjecture that there exist non-Desarguesian projective planes of all orders 2H,if n is greater than 3. With the advent of binary semifields, the gap between necessary and sufficient conditions on the possible orders of semifields has disappeared. TABLE OF CONTENTS I. INTRODUCTION. seccccccccccceccceccteccctecccceccccsvcecccescesees TI. SEMIFIELDS AND PREjSEMIPIELDS .cccccccsccccenscevesecessbssecess 2ol. Definition of semifield. 2.2. Examples. 2.3. Pre+ semifields, 2.4, The additive group. 2.5. Vector space representation, 2.6. Nuclei. Til, PROJECTIVE PLANES, TSOTORY ccccccscccscccrcccsccsecesceseseces 3.1. Homogeneous coordinates. 3.2. Isotopes. 3.3. The significance of isotopy in geometry. 3.4. Isotopy of semifields. 3.5. Nonlinear isotopes. TV. NONSINGULAR HYPERCUBES sc cccceccccccccceccecctccsecteeccncecece 4,1. Hypercubes and their elementary operations. 4.2. Sums and products of hypercubes. 4.3. Nonsingular hyper+= cubes. 4.4, Pre«semifields represented as 3-cubes. 4.5. Semifields and equivalent pre-semifields. Vv. TRANSPOSE OF A PLANE ccc cneccsccennseccecntaeeneseseresecesseses 5.1. Transpose defined. 5.2. Dual defined. 5.3. Collin- eations. 5.4. Examples, Semifields of order 32. VI. SOME KNOWN SEMIFIELDS ..ccccevccecceuccctevcvecscccscctscececens 6.1. Orders which are excluded. 6.2. Semifields of order 16. 6.3. Early work of Dickson. 6.4, Twisted fields. 6.5. Sandler's construction. 7.1. Definition of weak nucleus. 7.2. Quadratic exten- sions. 7.3. Construction of semifields. 7.4. The four types. VIII. BINARY SEMIFIELDS ce escesecccnsccccccccsesccscarsseseesseesece 8.1. The binary pre-semifields. 8.2. The binary semifields. 8.3. Isotopes. 8.4. Automorphisms. 8.5. Autotopisms. REFERENCES . ace vccc ct ese cece eee eee reece neces reese ease heer ssesebees 1 y LO 23 38 43 50 58 66 I. INTRODUCTION In this paper, the term semifield is used to describe an algebraic system which satisfies all properties of a field except for the commutativity and associativity of multiplication. Semifields are of special interest today because the projective planes constructed from them have rather remarkable properties. It is easy to show that the order of a proper semifield, i.e., a finite semifield which is not a field, must be p", where p isa prime, n is an integer greater than 2, and p" is greater than 8. But the question whether such systems exist for all p"™ meeting these requirements has been in doubt for many years. Various constructions of proper semifields have appeared in the Literature; the earliest works in this area gave many constructions for odd primes p and eventually the existence of proper semifields for all possible orders p™ was known, as long as p was odd. Constructions appeared later for the case 22, but in every case the exponent n was composite. Therefore, the existence of proper semi- fields of order 2%, where q is a prime number greater than 3, was still in doubt. In fact, no projective planes of these orders 2% were known, except for the Desarguesian planes derived from a field. But this question is settled in Section VIII of the present paper, where a class of proper semifields including all of the missing orders is exhibited. ee This paper is essentially self-contained; the theory of semifields is developed from the beginning, requiring no previous familiarity with the subject on the part of the reader. Section II contains a review of the basic definitions and theory of semifields, along with illustra- tions of some interesting semifields of order 16. In Section III, we describe a "homogeneous" notation for represent~ ing points and lines of an arbitrary projective plane in terms of its _ ternary ring. The concept of isotopy is generalized to apply to arbitrary ternary rings, and a simple method for mechanically con- structing all ternary rings isotopic to a given one is presented. Some known theorems about collineations of planes, and of semifield planes in particular, are proved concisely using the homogeneous notation. Finally, the question of whether a nonlinear "isotopy" can yield new semifields is considered. | Cubical arrays of numbers, of arbitrary finite dimension, ‘are the subject of Section IV. First, operations of transposition and muilti-~ plication are discussed. Then the notion of a nonsingular hypercube is introduced. Semifields are shown to be equivalent to a certain type of 3-dimensional cubical array, and the projective planes coordinatized by semifields are in 1-1 correspondence with equivalence classes of nonsingular 3-cubes. By transposition of a 3-cube, up to five new projective planes can be constructed from a single semifield plane. This construction is the topic of Section V. By exhibiting all of the semifield planes of order 32, with their interrelations and collineation groups, we give examples of transposed planes. 3. The finite semifields which have appeared in the literature are surveyed in Section VI, including a discussion of all semifields of order 16; of some commutative semifields due to Dickson; of the twisted fields due to Albert; and of seminuclear extensions due to Sandler. A general class of quadratic extensions is considered in Section VIL, in which the semifield is a vector space of dimension 2 over a so-called weak nucleus F. In particular, all quadratic extensions of F, for which F is equal to any two of the nuclei, are con- structed, thus generalizing a result due to Hughes and Kleinfeld. The last section deals with binary semifields, a new class of semifields of orders 2°, where n#3 and n #2", It is shown that many of these systems of the same order are isotopic. Some of these systems having m automorphisms are exhibited, and the autotopism groups are partially determined. The symbol "§" will be used throughout this paper to mean:: "This completes the proof of the theorem," or “This is all the proof of the theorem which will be given here." The most interesting results of this paper are found perhaps in theorems 3.3.1, 3.362, 363.4, 4.4.2, 4.5.2, 5.1.1, 5.2.1, 7.2.1, Teel, 8.201, 8.4.2, and 8.5.1. II, SEMIFIELDS AND PRE-SEMIFIELDS, We are concerned with a certain type of algebraic system, called a semifield. Such a system has several names in the literature, where it is called, for example, a "nonassociative division ring" or a "distributive quasifield." Since these terms are somewhat lengthy, and since we make frequent reference to such systems in this paper, the more convenient name semifield will be used. 21. Definition of semifield. A finite semifield S is a finite algebraic system containing at least two elements; S possesses two binary operations, addition and multiplication, which satisfy the following axioms, Al. Addition is a group, with identity element 0. Ae. If ab=0, then either a=0O or b=0O. A3. alb+ec)=ab+ac; (at bie = ac + be. Al. There is an element 1 in S such that la = al = Oe Throughout this paper, the term semifield will aliways be used to denote a finite semifield. The definition given here would actually be insufficient to define infinite semifields, for the stronger condition that the equations ax = b and ya=bd are uniquely solvable for x,y would necessarily replace axiom A2. Notice that a semifield is much like a field, except that multiplication of nonzero elements is required to be merely a loop instead of a group. De Every field is a semifield; the term proper semifield will mean © a semifield which is not a field; i.e., there exist elements a,b,c such that (ab)c # a(bc) in a proper semifield. 2.2. Examples. The following remarkable system V is a proper semi- field with 16 elements: Let F be the field GF(4), so that F has the elements 0, 1, ww? = l1+w. The elements of V are of the form ut+odv, where u,v c« F, Addition is defined in an obvious way: (a+ Av) + (x + Ay) = (utv) + A(xty) , (2.1) using the addition of F. Multiplication is also defined in terms of the multiplication and addition of F, using the following rule: (u + Av)(x + Ay) = (ux + v@y) + a(vx + u%y + vy?), (2.2) It is clear that F is embedded in V, and also that Al, A3, and A4 hold. To demonstrate A2, suppose that (u + Av)(x + dv) = 0, Then, in particular, ux + v?y = 0, so that, if neither of the original factors is zero, there is a nonzero element ze F, such that x= zv?, y = au. Then vx + uty + vey? = gv3 + gu3 + z2u%v? = 0. But this is impossible in F, unless u =v = 0. yY is certainly a proper semifield, since V is not commutative. But there is still a good deal of associativity present in V; in fact (ab)e = a(bc) if any two of a,b,c are in F, The remarkable property of V is that it possesses 6 automor- phisms, while the field of 16 elements possesses only 4. The automorphisms Oj; are given by: (u + Av)oy; = uw + dwtyd for i = 0,1,2; j = 1,2. (No other semifield of order 16 has as many automorphisms.) V is also anti-isomorphic to itself, under the mapping (u + Av)t = u+ Av?. Other examples of semifields which have properties in comon with V. will be discussed later. We will remark here, however, that if we had defined multiplication by the rule (u + Av)(x + Ay) = (ux + wv2y) + A(vx + vey) (2.3) rather than as in (2.1), we would have obtained another proper semi- field containing F; and this system has the stronger associativity property that (ab)e = a(bc), whenever any one of a,b,c is in F. Let us call the latter system W. We will see in Section 6.1 that a proper semifield must contain at least 16 elements. 2.3. Pre-~semifields. We say the system S is a pre-semifield if it - satisfies all the axioms for a semifield, except possibly Al; i.e., it need not have a multiplicative identity. A simple example of a pre-semifield can be derived from a field F which has more than one automorphism. In fact, let 95 be an automorphism, not the identity, and define x ° y = (xy)°. Then (F,+,0) is a pre-semifield, and it has no identity; for19°1=1 implies that 1 must be the identity if any exists, yet lo y= yo #y for some y. q. We will see in Section IV that a finite pre-semifield can be thought of as a three-dimensional array of integers, and that a semifield can be constructed from a pre-semifield in several ways. 2.4. The additive group. It is easy to show that the additive group of a pre-semifield S must be commutative. By the distributive Laws, (ac + ad) + (be + bd) = (a + b)(c + da) = (ac + de) + (ad + bd), Therefore, by Al, ad + be = be + ad, and any elements which can be written as products commute under addition. But by A2 and finiteness, any element of S can be written as a product; therefore, the additive group is Abelian. Another simple argument shows that the additive group is elementary Abelian. In fact, let a#0O, and let p be the additive order of a, Then p must be a prime number, since (na)(ma) = (nm)a for integers n,m. The fact that every nonzero element has prime order suffices to show that the group is elementary Abelian, and that all nonzero elements have the same prime order p. This number p is called the charac-. teristic of the pre-semifield. 2.5. Vector space representation. Let S be a pre-semifield, and let F be the field GF(p), where p is the characteristic of 8S. Then we can consider the elements of F to be "scalars," and S is a vector space over F. In particular, S must have p”™ elements, where n is the dimension of S over F. The simple observations just made in the preceding paragraph are surprisingly useful, since many of the concepts of semifields and 8. pre-semifields are fruitfully translated into vector space terminology. This will be exploited further in Section IV. We can, for example, rephrase the distributive laws as follows: Let a be a nonzero element of 5S; we define the functions Le and R. as follows: bL,. = ab; bR, = ba. (2.4) Then the distributive laws A3 are equivalent to the statement that Li. and Ro are linear transformations of the vector space into itself. Furthermore, the axiom A2 states that these transformations are all nonsingular, if a#0. Therefore, L, and R, can be represented as nonsingular matrices, with elements in GF(p). 2.6. Nuclei. Various special subsystems are defined for a semifield S, indicating degrees of associativity. The most important of these are the following: The left nucleus N,: { x | (xa)b a: x(ab), a,b € S}. The middle nucleus Nj: {x | (ax)b = a(xb), a,b c S}. a(bx), a,b e S}. The right nucleus N,,: { x | (ab)x The nucleus N is the intersection of the left, middle, and right nuclei. The fiela GF(p), where p is the characteristic of Sy is obviously always part of the nucleus. In many cases, the nuclei are all trivial, i.e., equal to GF(p). This is the case for the system V in Section 2.2. But the system W described in that same section has nucleus F = GF(k). 9. It is easy to verify that each of the nuclei is actually a field. Furthermore, S is a vector space over any of its nuclei: it is a left vector space over N,, N and N; it is a right vector space m? over Ne? N and WN. The operations Le and Re defined by y? equation (2.4) are not necessarily linear transformations over the nuclei; but Ro is a linear transformation over Nos and Le is a linear transformation over Nh when S is regarded as a left, right vector space, respectively. 10. III, PROJECTIVE PLANES, ISOTOPY. Perhaps the major application of semifields today is for the construction of combinatorial designs, and of projective planes in particular. Every proper semifield determines a non-Desarguesian projective plane. In this section we discuss projective planes, and the question whether two semifields coordinatize the same plane. 3.1. Homogeneous coordinates. Let # be a projective plane, and let T be a ternary ring coordinatizing 1, as in [1]. We write the ternary operation a*bec, and write as usual a*boO = ab, a-rlec = atc, It. is convenient to introduce a new type of coordinate system, for simplicity in notation, as follows: Homogeneous Notation Notation in [1] Points; (0,0,1) ( » ) (O,l,a), aet (a) (l,a,b), abet (a,b) Lines: [o,o0,1] L{ @ ) [oO,l,a], aeT x+a=0 [l,a,b], a,bdeT y=x*aobd The principal feature of this notation is that the point (x, x, 2x.) is incident with the line [yy s¥os73] if and only if Y%, = X,*°¥, 0 XY5° (3.1) ll. Since ¥, and x} must equal OQ or 1, the meaning of the "multiplications" ¥4%, and x75 is clear in this relation. The fact that incidence can be expressed in a single formula means that many special cases can often be eliminated, when carrying out proofs. The symmetry of this definition also makes the duality relation clear. We let A:B denote the line joining points A and B, and let LAM denote the point common to lines L and M. Furthermore, we define the elements a and “a by the equations (a¥) +a =0, a+ (va) = 0. (3.2) The following formulas can be easily verified: “(av) = (Ya)¥ = a (3.3) [0,0,1] = (0,0,1):(0,1,0) (0,0,1) = [0,0,1] n[0,1,0] [0,1,%a] = (0,0,1):(1,a,b) (0,1,a) = [0,0,1]A[1,2,b] [l,a,b] = (0,1,a):(1,0,b) (1,a,b) = [0,1,%a]a[1,0,b] 3.2. Isotopes. Let T and T, be ternary rings. An isotopism 1 from T, onto T is a set of three functions (F,G,H), each being i-~1 correspondences from T_, to T, such that 1 (O)H Hu 0, L'§ (asboc)H = (aF)-(bG)o(cH), for all a,b,ce T. Theorem 3.2.1. Let (F,G,H) be an isotopism from T, to T. Then H = Fy = G@ (3.5) where ¢ = Lins y= Rig? aadwhere L and R denote left and right multiplication in the ring T. le. Proof: Set c= 0 in formula (3.4) to obtain (ab)H = (aF)(vG). In particular, aH = (aF)(1G) = aFR, >, = aFv, bH = (1F)(bG) = bGL, p = bGd. i Theorem 3.2.2. Let 7 be a ternary ring with n elements. The number of nonisomorphic ternary rings isotopic to T is at most (n=1)2, Proof: Let T and T, be ternary rings isotopic to T, under the functions (F_,G ,H ): T +T and (F ,G ,H ): T > T. Let the 11.iul 1 2° 2 2 2 ternary operation of R, be denoted by (a,b,c) and let the ternary 1 operation for R, be denoted by [a,b,c]. We will show that if 2 1F, = 1F, =y, and if 1G, = 1G, =z, then T, and T. are “dL 2 2 1 2 isomorphic. Since there are (n-1) choices for IF, and (n-1) independent choices for 1G, there will be at most (n-1)? non- isomorphic ternary rings, as claimed. By Theorem 3.2.1, we have H u Q ee a mY 1 Fy y = G, o = Hos where $6 = Lys v= Ro Hence, FURS? = 66,7) = HH? = a. Now a is the required isomorphism, since [aa,ba,ca] = (aoF,*baG,ocall,, HS" = (aF, *bGocH, )HT'a= (a,b,c)a. | We note that the limit (n-1)* is best possible; there is a ternary ring with 32 elements which has 31% distinct isotopic ternary rings. 13. (See Section V.) But if T is a field, we have the other extreme where all isotopic rings are isomorphic to T. Theorem 3.2.3. Let T be a ternary ring, and let y,z be nonzero elements of T. Let $= Lys ¥=R,, and F = y!, G = o-!, and let qT) be the system consisting of the elements of T with a new ternary operation defined as follows: (a,b,c) = aF* bGoc. | (3.6) Then T is also a ternary ring, having the identity element yz. 1 Ty is isotopic to T, and furthermore all ternary rings isotopic to T can be constructed in this way (up to isomorphism). Proof: Note that F and G are well-defined since y and 2 are nonzero. The latter part of this theorem follows from the pre- ceding theorem; we must show only that (a,b,c) satisfies the require- ments for a ternary ring. I. (0,b,c) = O*bGoe = c = aF*Ooc = (a,0,c). . II. (yz,b,0) = y*bGoO = y(bG) = dG = b. (a,yz,0) = aF*zoO = (aF)2 = aFy = a. III. We can solve (a,b,x) = c uniquely for x, since aFebGox = ¢c is uniquely solvable for x. IV. We can solve (x,b, ,¢,) = (xb, 4¢p) uniquely for x, if Gee, = xF*b,Goe 1 2 2 is uniquely solvable by 7 b,, since xF+d, for xf. V. Finally, we can solve (a, Wx) = Cys (a, sw 4x) =c¢, uniquely for (w,x), if a, # a9 since we can solve a FewGox * Gy a,,F* wGox = ¢, uniquely for (wG, x). | 1k. Theorem 3.2.3 is essentially a converse to theorem 3.2.1, for it says that the relations (0)H =0 and H = Fy = Go are sufficient to construct a new ternary operation; no stronger condition can be derived from general isotopy. Theorem 3.2.3 also provides a convenient way to calculate all ternary rings isotopic to a given one. An isotopism where H is the identity is called a "principal isotope." 3.3. The significance of isotopy in geometry. An isomorphism a between projective planes is a 1-1 correspondence between points and lines which preserves incidence; i.e., point P is on line L if and only if point Pa is on line I®, An automorphism of a plane is commonly called a collineation. Theorem 3.3.1. Let m and t' be projective planes and let a be an isomorphism from tw' onto t- such that (0,0,1)a = (0,0,1) (O,1,0)a = (0,1,0) (1,0,0)a = (1,0,0). Then the ternary rings of 1m and nt' are isotopic. Proof: [0,0,1Ja = (0,0,1)0:(0,1,0)o0 = [0,0,1]. Therefore, (0,1,a)a0 lies on [0,0,1], and there must be a 1-1 correspondence G such that (O,l,a)o = (0,1,aG). . (3.7) Similarly we find [0,1,0Jo = [0,1,0], [1,0,0]o = [1,0,0]; hence there are 1~1 correspondences H,F such that (1,0,b)a (1,0,bH). (3.8) (1,a,0)a (1,aF,0). 15. We can now ecaleulate the images of all lines: [O,1,vala {l,a,bJa = Finally, we find the Now we can derive i.e. (L,a,b)o (1,X5 .X3) & [2 yos¥3] - (1,X5 x3) € [lsyosyg]a *3 i! R the desired law: _ (0,0,Ll)a:(1,a,0)a (O0,1,a)a:(1,0,b)0 = [0,1,%a]on[1,0,bJo = (1,aF,bH). [0,1,%(aF) J (3.9) {l,aG,bi]. image of every point: (3.10) -(3.11) xH = 3 X_Fry ,Goy .H, and this is precisely the relation used for isotopy. | Theorem 3.3.2. (Converse of Theorem 3.3.1). isotopy from a ternary ring T' to T, Let (F,G,H) be an and let ™',™ be the corresponding planes. Define © by equations (3.7)-(3.10); then a is an isomorphism between w' and 7, Proof: Since a directly from the law consider. I. II. Til. (1,X5 4X3) (14%5 9x3) 0 (0x5 5x) (0X5 5%) (0X5 5X3) (0,x, 9X3) 01 is 1-1 and onto, of isotopy, there € [0s¥osy3] <> € [0,yosy3]la e (Msyo.y3] € [lsyosygla <> € [0,yosy3] +> - € [0,Yos¥g]a We define autotopism in an obvious ternary ring onto itself. If (F,G,H) and since (3.11) holds are only a few cases to Yo * i and y= Xo Yo = il and ygVP= xP. X, = 1 and %3 = Vo Xo =] and XG = YoGe Xp = 0 or Yo = 0 X70 or yo = 0. ] way, as an isotopism of a and (F',G',H') are autotopisms, 16. (a*boc)HH' = (aF+bGocH)H' = aFF'-bGG'ocHH', so we define the product of two autotopisms as (F,G,H)(F',G',H') = (FF',GG',HH'), | (3.12) An automorphism is the special case (F,F,F) of an autotopism where all three permutations are equal. Corollary 3.3.3. All isotopic ternary rings coordinatize the same projective plane. The collineations of a projective plane which fix (0,0,1), (0,1,0), and (1,0,0) form a group isomorphic to the group of autotopisms of the ternary ring. | Theorem 3.3.4. Let T be a ternary ring with n elements, and let h be the order of the autotopism group of T. Then (n-1)2 (3.13) where T' ranges over all nonisomorphic ternary rings isotopic to T, and where k(T') is the number of automorphisms of T'. Proof: Let y,z range over the nonzero elements of T, and consider the (n-1)? ternary rings constructed in Theorem 3.2.3. If T' is any of these ternary rings, we will show that there are exactly h/k(T') ternary rings of the set which are isomorphic to T', and this will prove formula (3.13). We need only show that h/k(T) of the ternary rings are isomorphic to T, because the autotopism group of T' is conjugate to the autotopism group of T and therefore has the same order, and because 17. the (n-1)2 ternary rings formed from T' are isomorphic in some order to the (n=-1)* ternary rings formed from T (using Theorem 3.2.2, since the rings are determined by 1F and 1G). Let o be an isomorphism from T to the ring o(R yu 7 at). There are k(T) such isomorphisms. ‘Then (asboe)o = (aa,ba,ca) = aakZ+baly loca, i.e., (oR? ,oL5' a) is an autotopism. By Theorem 3.2.1, every auto- topism is of this form, and@ defines y and 2; therefore, if r of the pairs y,z2 yield isomorphic rings, there are h = rk(T) auto- topisms. | 3.4. Isotopy of semifields. A semifield is a particular type of ternary ring, where a*boc = ab + c and axionis Al-Al hold. We now specialize the material of the preceding sections to the case of semifields. Theorem 3.4.1. Let S be a semifield of characteristic p. All ternary rings isotopic to S are semifields. (F,G,H) is an isotopism from 5' to S if and only if F,G, and H are nonsingular linear transformations from S' to S over GF(p), satisfying the condition (ab)H = (aF)(bG). (3.14) Proof: Suppose S' is isotopic to S, and (a*boc)H = (aF)(bG)+cH. Then (a*joc)H = (aF)(1G) + cH = aH + cH, so H is an isomorphism between the addition of S' and the addition of S, i.e., H is a nonsingular linear transformation over GF(p). Now H = Fw = Go 18. by Theorem 3.2.1, where and 4, being functions of left and right multiplication, are nonsingular linear transformations over GF(p) of S into itself; hence F and G are also nonsingular Linear transformations. Properties Al through Al are now verified immediately, as is the converse portion of the theorem. | Theorem 3.4.2. Every collineation of a plane coordinatized by a proper semifield fixes (0,0,1) ana [0,0,1]. Proof: This theorem is well known, but its proof requires the development of more geometrical tools then are appropriate here. Proofs may be found in [2] and [3]. f The previous theorem holds primarily because any semifield already has a great number of collineations, and if it were permitted to move the point (0,0,1) or [0,0,1] it would have so many more collineations, it would become Desarguesian. The standard collineations, holding in any semifield plane, are the translations t(h,k) and the generalized shears o(h,k), defined for all h,k in the semifield as follows: (x1 Xo 9X3)1 (hk) = (x1 .Xptx,h X9tx,k) [y 5 s¥ps¥q]}t(h,k) = [Yq so s¥ g-hy ty, k] (3.15) (x, 5X5 ox,)o(h,k) = (xy 2X5 oXqtxyh*x, k) [yy s¥pey,]o(hok) = [y,s¥gty hey zty,k] The proof that these are collineations is very simple with homogeneous coordinates: ¥y%3 = Xoo + 173 19, if and only if yy(x3+x,k) = (xptx,h)y, + x,(yg-hyyty,&) if and only if Yy(x3txh+x,k) = Xolyoty zh) + x, (yatyyk), remembering that x7 and Y, are restricted to be 0 or 1. The following relations are easily computed, using the formulas already derived; here o(F,G,H) represents a collineation correspond- ing to an autotopism. t(0,k) = of(0,k) t(h,k)t(h' yk") = t(hth'k+k') " o(h,k)o(h',k') o(hth! ,k+k') a(F,G,H)a(F',G',H') = a(FF',GG' HH) (3.16) t(h,k)~!o(2,m)t(h,k) = o(2,m-h£) ul o(h,k)~!1(2,m)o(h,k) t(£,m+2h) a(F,G,H)"!t(h,k)a(F,G,H) = t(hF,kH) a( F,G,H)~'o(n,k)o(F,G,H) o(hG,kH) Theorem 3.4.3. Two semifields coordinatize the same plane if and only . if they are isotopic. Proof: If 8 is an isomorphism between two semifield planes, then (0,0,1)8 = (0,0,1) and [0,0,1]g = [0,0,1] because this point and line are characterized by Theorem 3.4.2. Henee (0,1,0)8 = (0,1l,a), and (1,0,0)8 = (1,b,c). Let a = Bo(-a,0)t(=b,ba-c). Then (0,0,1)0 = (0,0,1); (1,0,0)a = (1,0,0); (0,1,0)a = (0,1,0); and Theorem 3.3.1 applies. The converse is part of Corollary 3.3.3. | 20. Theorem 3.4.4. Let G be the collineation group of a semifield plane; let T be the subgroup of translations, 5S the subgroup of generalized shears, H the subgroup corresponding to autotopisms , and A the elementary Abelian additive group of the semifield. Then we have the following normal series for G: H TWTtnsSa Ta Tvs Tver G. The quotient groups are equal respectively to A, A, A, and H. Proof: This follows from formulas (3.16), and the fact that TUSwH = G as proved in the preceding theorem. { Theorems 3.4.3 and 3.4.4 are well known; they indicate how important isotopy is when considering semifield planes. The use of homogeneous coordinates simplifies and clarifies previous proofs of these theorems. 3.5. Nonlinear isotopes. One might pose an interesting problem here, concerning whether linearity of F,G,H is really a necessary condition for constructing semifields. Suppose we have a semifield S and we have defined a new multiplication * on S by the formula (a®>)H = (aF)(bG). We require that F,G,H be permutations of 3S, and that (atb)*e = aXe + b¥e,. ex¥(atb) = cxa + cx#b, (3.17) Question: Does this imply that F,G,H are linear? al. The following theorem does not settle this question by any means, but it does provide some insight into the matter. We say an element x eS is nonecentral if xy = yx implies that y=r+sx, r,s e€ GF(p). Theorem 3.5.1. If S has characteristic 2, and if every element of S - GF(2) is noncentral, then the functions F,G,H of (3.17) must be linear. Proof: The conditions imply that aH™! = aF7!,1G™! = 1F7)yac7!, so we can write H = PF = QG where P and Q are linear. Thus (aH)(bH) = (aP x bQ)H. (3.18) Let x be such that xH = 1. Then since (aH)(xH) = (xH)(aH), equation (3.18) shows us that aPxyxQ = xPsaQ, for all ae S. Adding aPxaQ to each side, we obtain aP » (atx)Q = (atx)P « aQ; hence (aH)((atx)H) = ((atx)H) (aH), i.e., aH commutes with (atx)H. This is the relation which we will use in order to show that H is linear. We can at least prove that (0)H = 0, since (0H)(aH) = (OPyaQ)H = (O)H for all a, ‘Therefore x #0. We will now show that (a+x)H = aH + xH = eH +1, for all a, (3.19) If a=0O orif a= x, this is obviously true (using the fact that x +x = 0). Otherwise aH is noncentral, by hypothesis. By the definition, this implies that (atx)H = 0 or 1 or aH or aH +1. 22. The first three possibilities are clearly impossible, so (3.19) is established, Since H = Gd, where $ ‘is linear, we have (atx)G = (atx)Ho7) = alow! + xHom! = aG + xG. Finally, then, let a,b be arbitrary nonzero elements of §$; define c,d such that a= cxx, b = cxd. Then Lt] i} (a+b)H (cx(x+d) )H (cF)((x+a)G) = (¢cF)(xG + dG) H (cF)(xG) + (cF)(adG) = aH + dH. Thus H is linear and therefore so are F and G. | Remark. The above theorem is not trivial, for there exist systems in which the hypothesis is satisfied (e.g., V in Section 2.2). Furthermore, some hypothesis is necessary for the theorem, because there are examples in which F,G,H are not linear. Such an example is the field S = GF(8). Let x bea primitive element with minimum polynomial x3+x24+1; then define H by ( xi (moa x3+x2+1) )H = xl (mod x3+x+1). If we write (axzb)H = (aH)(bH), the equations (3.17) are satisfied, but (1 +x*)H=1l+x #1 + x? = 1H + x7H, Such nonlinear isotopies do not have geometric significance, however, according to Theorem 3.4.1, and they will not be considered further in this paper. 23. Iv. NONSINGULAR HYPERCUBES. In this section we turn to another way of looking at semifields and their isotopisms, where we think of 3-dimensional matrices. The. 3-dimensional representation allows us to see several symmetries in the situation which are not otherwise apparent; and it also allows us to work with pre-semifields, when it is convenient. 4.1. Hypercubes and their elementary operations. An m-dimensional hypercube A, m21, is an array of n™ elements belonging to a field; the elements are denoted by Ais.eer where there are m sub« scripts, and each subscript varies from 1 to n. We consider n to be a fixed integer- throughout the entire discussion. It is easy to devise extremely cunbersone notations for such systems, so an attempt will be made to keep the notation as simple as possible. Let o be a permutation on the elements Ly2yeeete Then A? will represent the m-cube A with subscripts permuted by Os ‘i.e., the ko-th subscript of A° is the kth subscript of A. This is a generalization of the concept of transposition of matrices; if A is all2), In the 3-dimensional case, if 2), at a matrix (m Az (A then if B = A(!23) = (By ,,) we have Bia = A for ijk)? ijk jki all i,j,k. We also have the general law (A°)™ = Ao, 2h, If m?>i1, we form (m-1)-dimensional subcubes of an mecube A, denoted by alt], as Tollows: B= alt] = (3 ) <-> B = ijeeer ~ Atij...r’ (4.1) ij...r Here 1< +t <n. Additional subcubes which hold other positions fixed can be formed by combining the operations A? ana alt] described here; actually all conceivable subcubes can be obtained in this manner, 4.2. Sums and products of hypercubes. The sum of two m-cubes A and B is simply defined as the m-cube consisting of the sums of the components: C=A+B* (Cyn) A =C (4.2) a + ss ss LjeeeDl Bi jeer Ljeee? The products of a p-cube B times an m-cube A, yielding an (m+p-2)-cube, can be defined in the same way as the tensor dot product of tensors, as follows: 1 ) Ci dkeeer } Be edt itk...re Actually, n of such products can be defined, and we write in general Saez (c )— C=B i... jk...qree.8 (4.3) Cs ad ReeeQheeeS = ) Be gthi...jtre..s where "i...j" represents -1 subscripts, "r...s" represents n-f. ] x Notice that alt] =B* A, where B is the l-cube (B,) = (°;,). . 1 2 We also see that if A,B,C are matrices, Bx A= BA, CxAS= act, 25 expressing the familiar fact that premultiplication corresponds to operating on rows of a matrix, postmultiplication corresponds to operating on columns. The associative law for matrix multiplication, (Ba)cl = B(aC"), can be written in the form C x (B x A) =B x (c x A). This relation is a special case of a general rule for products of hypercubes. Theorem 4.2.1. (Generalized associative law). If u<v, and if the dimension of C is f+ 2, then c¥ia ka) = Bs’ (c Ba) (4.4) Proof: Let the dimensions of A and B be m and p, respectively. The general element D of Lowe d Keo Glee eSWeeeXVouedZ the product C X (B x A), where i...j represents u-l subscripts, k...q represents f+1 subscripts, r...s respresents v-u-1 sub= scripts, w...x represents p-l subscripts, and y...z represents m—v subseripts, is, by definition, ) Ck. atl ) By. ..xbAL.. jtr...shy...2)° The same general element of the right-hand product is A 2 By. sxe : Cy gh4i.. jhr.sesty...2)? and these are clearly equal. [| 26. Corollary 4.2.2. Let m be the dimension of A, and let By Bose eB, be matrices. Then the m multiplications +B, x A can be performed on A in any order, i.e., if gf is a permutation, By 1 (By 2 ‘a (B, x A)e**)) 18 28 m8 (4.5) Proof; If the dimension is 2, then f in Theorem 4.2.1 is always zero, so this result follows by repeated application of that theorem. | | We will write [B, By,++-B,] x A to denote the multiplication operations expressed in equation (4.5). The following law generalizes the matrix equation (ca)? = atc?, Theorem 4.2.3. If C isa matrix, and if o is a permutation, (c¥ayy = cea | | (4.6) Proof; Let the subscripts be ij sigseeeil- The desired result follows from the formulas (Ae), =A. . ijigesed, t1o*20°** mo u (c x A) = Jc aA, i i eoel i t i eeoeolk ti esel . “12 mn t nt 1 uel atl *m (c x a’) = )c. iA i i easel i t i eoel ti eeol s yr ott ttm t “uo “lo (u-1)o “(Cutie mo I aT. Several other associative laws can be formulated to include the cases not meeting the hypothesis of Theorem 4.2.1; i.e., when we are . u ¥ given a product Cx (B x A) such that wu > Vand that ut2-dim C < v. But only one of these is of concern to us here: Theorem 4.2.4. [Cy sCoseeesC,] * (1B, Bogee-5B.] * A) = [CB] ,CoB,,++-C,B] x As | (4.7) Proof: In lieu of Theorems 4.2.1 and 4.2.3, it suffices to show 1 1 1 . that (C x (B x A) = (CB) x A when C,B are matrices. The general element dD, of the left hand side is eae ) C0 Bendnj. eer)? ‘on the right hand side it is c,B JA ; } 0 ih nid ties el l and, these are equal. 4.3. Nonsingular hypercubes. The concept of a nonsingular matrix ean be generalized to m-cubes in a significant way. The definition proceeds inductively: We say that a vector, or l-cube, A is singular if and only if A= 0. For m> 1, we say that an mcube A is singular if and only if there exists a nonsingular vector B such 1 that Bx A is singular. 28. Another way to state this definition is that an m-cube A is nonsingular if the following condition is satisfied: "x al tlex al2d4 coe + x alm] is singular implies that In other words, any nonzero linear combination of the subcubes alt] must be nonsingular. This definition certainly includes the ordinary definition of a nonsingular matrix, for in the special case m= 2 it says that the rows of A are linearly independent. Theorem 4.3.1. Let A be an m-cube, and let o bea permutation of {1,2,...,m}. Then A is nonsingular if and only if A° is nonsingular. Proof: We use induction on m. For m= 1 it is trivial, and for m= 2 it is a special case of the theorem that “row rank equals colum rank of a matrix." We assume, therefore, that m > 2. Without ioss of generality we may assume that o is of the form (12) or (23°°*m), since these two permutations generate all of the m! permutations. If o = (12), we argue as follows, where B and C ‘genote vectors: A is nonsingular <« (C#0+C . A is nonsingular) — (C#0>(BF#O>% Bx(C x A) is nonsingular)) since m> 2 > (B#07 (C#O> Bx(C x A) is nonsingular) ) 1 2 (Bp #07 (c #0 Cx(B x A) is nonsingular)) by Th. 4.2.1 —_ 2 > (Bp #07 B* A is nonsingular) 1 1 (12) 2 +> (B #0> Bx AS is nonsingular) since B* A =B x A _ AS is nonsingular. 29. On the other hand if o = (23**'m), let +t = (12°°*(m-1))., 1 1 Then since (B x A)™ = B x (A%) we argue as follows: . : 1 A is nonsingular «* (B #07 8x A _ is nonsingular) 1 -— (B #07 (Bx A)t is nonsingular) by induction 1 o . . «> (B #078 x (A°) is nonsingular) «+ A is nonsingular. | Theorem 4.3.2. Let A be an m-cube and let Cy sCoseeeyC, be nonsingular matrices. Then A is nonsingular if and only if [C, Ca see0 9] x A is nonsingular. Proof: For m=1 this is merely the definition of a singular matrix. For m> 1, it suffices to show that A is singular if, and only if, C x A is singular because of Theorem 4.2.3 and Theorem 4.3.1. xh 2 1 Cx A is nonsingular «* (B #07 Bx (C x A) is nonsingular) Ke 1 <> (B #07 Cx (B x A) is nonsingular) by Th. 4.2.1 1 «+ (B #07 B* A is nonsingular). by induction ++ A is nonsingular. | Finally, we define an equivalence relation between m-cubes as follows: A zB if and only if A= [C, Cases] x B . (4.8) 30. for nonsingular matrices CC aseee sl - By Theorem 4.2.4 this is m an equivalence relation, and by the preceding theorem this preserves singularity. 4.4, Pre-semifields represented as 3-cubes. We now specialize to the case m= 3. Let S be a pre-semifield of characteristic p. Accord- ing to Section 2.5, S is a vector space over F = GF(p). Let {X) 2X 0006 5%) be a basis of S over F. We can write the multipli- cation in terms of the basis elements: x, x, = )} A (4.9) . X. 6 ij x ijk k This gives us a 3+cube, A, with entries in F. A is called a cube corresponding to 5. The multiplication in S is completely determined by the products of the basis elements, according to the distributive laws since 0 eure! ex) = } } bic sxx, =) } ) arin a (4.10) Theorem 4.4.1. A cube corresponding to a pre-semifield is nonsingular. Conversely, if A is a nonsingular 3-cube, we can define a pre- semifield S by equation (4.10), with the x; being formal basis elements. Proof: Because of our other observations, we need show only that mitiplication defined by equation (4.10) satisfies axiom A2 if and only if A is nonsingular. In the proof of Theorem 4.3.1, we have shown that A is nonsingular if and only if 31. 1 B#0 ana c#0 > cd (Bia) #0 (in the case m= 3). ] 1 But Cx (Bx A) is the vector D = (D,) where Dy, = ) 5. ) Cy Ask} r s therefore, this is precisely the condition that a#O and b#0 + ab#oO. J] We observe that a nonsingular 4ecube would correspond to an algebraic system with a ternary multiplication abe satisfying " In general, three distributive laws, and with no "zero-divisors.' @ nonsingular m-cube will lead to an (m-1)-ary operation. If we change to a different basis ty, seeesyt of S over F, we have y,* y C5 5X5. where Ce (C; 5) is a nonsingular matrix. J This introduces a corresponding change in A; let the new cube be. B. Then. 43 . () C3) ( Cis¥*s 7 y } ) Ci ish rsk* r Ss rek «JY J oc, arly . rskt ir’js rsk "kt", Now Bs sk is the coefficient of vis hence Be [c,c,ct]xa, (4.12) where -T denotes inverse transpose. Thus B =A. 3e. Let us now consider what would happen if we were to define a new multiplication on the elements of 5S: (ax b)H = (aF) (dG) (4.12) where F, G, H are arbitrary nonsingular linear transformations of S into itself. This gives us another pre-semifield S', which is said to be isotopic to S, Let B be a cube corresponding to the derived pre-semifield st. We may assume S' has the same basis {x sXyo00eX } as S, and that A is the cube for 3 with this basis. Consider F,G,H as matrices 3 i.e., x.F = JFox. (4.13) i , irr v = —=j = 3 _—se Then x, 0%, (Cx; F) (x, 6) )B (0 Ft) G5%5) 8 ~l = ( ) y “iSsetonete rst ~ d : d L Fir’ jst ok Ast ™" (4.14) Therefore B= [F,G,H7]«A. | (4.25) This proves the following fundamental theorem. Theorem 4.4.2. Let S and S' be pre-semifields, and let A,A' be any cubes corresponding to S and S‘', Then S is isotopic to St if and only if <A =A' (where equivalence is defined in equation (4.8)). 33. 4.5, Semifields and equivalent pre-semifields. The preceding discussion applies to semifields as special cases of pre-semifields. If S is a semifield, we can assume the basis is of the form {15% se00 9x Je The corresponding cube A then has a special property. We write Aree = alr] awre = (a6 232) [x] | (4.16) ARRYT = (aC 223) y[r], In other words, AT*#, A*T#, AX#r are the matrices (Ang g) (Agni) (A 5p) respectively. AT** is the matrix Ly of left m1tiplication by r x A*r® is the transpose of the matrix for right multiplication, . R,..« The following formulas are readily verified: *y ({P,G,H] x a)re* =F (GalwHT) +... + F (Gammon?) , ({F,¢,H] x a)er® = (Hastert) +... 4G) (Hammer), | (4.17) vi T ((F,G,H] * A) ##r H (Pawel ) * oe. + H_, (FA#IDG ). We say A is in standard form if Al#* = avl® =], Theorem 4.5.1. Let S be a semifield with basis {1,X5,0005X,}5 then the cube corresponding to S is nonsingular and in standard form; conversely, every nonsingular 3-cube in standard form yields a semifield, if multiplication is defined by (4.10). 3h. Proof: Because of Theorem 4.4.1, we must merely observe that standard form is equivalent to axiom Ak. But this is obvious, since standard form is nothing but the statement that Ly = R, =I, | Theorem 4.5.2. Let S,S' be semifields with corresponding cubes A,A'.. Then S and 8S! coordinatize the same projective plane if and only if A =A‘'. Semifield planes are in 1-1 correspondence with equivalence classes of nonsingular 3~cubes,. Proof: Apply Theorems 4.4.2 and 3.4.3. | Theorem 4.5.3. if A is a cube corresponding to a proper semifield S, the autotopism group of S is isomorphic to the group of all triples of matrices (F,G,H) such that [F,G,H]xA = A. (4.18) Proof: This follows from equation (4.15). |] We note that the same result holds for any 3-cube B, if B2A, even if B is not in standard form. | Now we turn to the question of constructing a semifield from a pre-semifield. This is merely a question of finding three nonsingular matrices such that [F,G,H] x A is in standard form. Such a con- struction can be done in several ways; for example: 1. Set G= (al*¥*)"], set BEG P A, set F= (pele)-T H=TI. 2. Set F=(avt*)-T, set BEF x A, set G = (Bl¥*)"],H =I, 3. Set H = (Ale#xj-T) ps alwa(qele)-T) Gg = 7, 4, Set H = (A#1}*)~1, Gg = q¥leT(qle#¥j"l, Fer, The proofs that these do the job are similar, using equations (4.17). We consider for example method 3: a a a a a ooo B ~1 10 eee O 111 112 lin 111 =#112 lin a a a a - 121122 *1on| | "2117212 ® oan] = e ° e e * e e e een * % * Therefore ([F,G,H] x A)i## = GA! xeq! =I; and ([F,G,H] x A)#!* = Hae] «p> = I. These methods can all be translated into algebraic terms; e.8., here is method 3 in this form: Theorem 4.5.4. Let (S,+,0) be a pre-semifield, and let ue S. Then if- we define a new multiplication » by the rule (a°u)* (vod) = aod | (4.39) we obtain a semifield (S,+,#) with uit ur. | Notice that if (S,+,°) is commutative, so is (S,+,#). Formula (4.19) is really an instance of equation (3.6) which is actually more general. A less symmetric way to write this formula can be read directly from method 3, namely u° (a * b) = (uo a)RU'o b, 36. which yields an isomorphic system with unit element u. Method 4 gives another isomorphic system, dual to this one. Thus, we can obtain semifields from pre-semifields in several ways; in each method we were able to leave F,G, or H equal to the identity. No matter which way we choose, the result is equivalent in the sense that the same projective plane will result. The extra degree of freedom we seem to have in this standardiza- tion process indicates that we should seek a "more standard" standard form, so that it might actually be a canonical form. One idea sug- gests itself immediately: we could require that A**l =I also. If S is a field with p* elements, it can always be put into the form 10 Ol Ale® = A2e* = O01 la $ if a is chosen such that x? + ax -1 is irreducible (mod p). The field GF(8) can be put into the forms 100 O10 O01 100 010 001 010 101 O01 or 010 dill Oil 001 O11 111 0Ol O11 110 where we adopt the convention of writing A!®*, A2#*, vee in this order. It is unelear whether this form is possible in general. These examples show that it is not a canonical form; but if it is true that all nonsingular cubes can be standardized in this way, it would be useful for eliminating cases when constructing the systems. 37. A way to utilize the extra freedom which has been very fruitful is to adjust A*#*, We can essentially perform any desired similar-~ ity transformation on A2ee while the standardization is taking place; therefore, in particular, if the characteristic equation of A*%* after standardization is irreducible, we can transform it into ‘a companion matrix and restandardize; this means we are taking the basis of the vector space to be of the form 1, x, x2, x(x*), x(x(x2)), etc. Operations of this kind can be exploited when all possible semifields of a given order are to be constructed [see reference 2]. 38. V. TRANSPOSE OF A PLANE. This section discusses an interesting relationship between some Planes coordinatized by semifields. The relationship is somewhat peculiar since it has algebraic significance but does not seem to have any readily seen geometric significance. 5.1. Transpose defined. Let a be a projective plane, let 5 be a semifield of coordinates for a, and let A be a cube corres~ ponding to 8. Let S, be the pre-semifield described by a‘23), and let So be a semifield constructed from 5) py isotopy. Then we define nt, the transpose of a, to be the plane coordinatized by Sie Theorem 5.1.1. - is uniquely defined. Proof: No matter which choice of S is made, the resulting A's will be equivalent, by Theorem 4.5.2. The 3-cube al23) is nonsingular, by Theorem 4.3.1. If A =B, then a(23)s p(23), because of Theorem 4.2.3. Therefore 23) is uniquely determined up to equivalence. Thus the plane tm is uniquely determined. | Corollary 5.1.2. (nT)T = 1. ] 39. 5.2, Dual defined. The dual ™ of a projective plane coordinatized by a semifield is well known to be determined by constructing the anti-isomorphic semifield, i.e., replacing ab by ba in the semi- field. (This is easy to see by considering the homogeneous coordinate notation.) This definition can be phrased in the same way as the definition for w!, vy using a‘l2) rather than al23), We may write wD = o(l2)) gl 2 (23), Theorem 5.2.1. The operations of dual and transpose generate a series of at most six planes, according to the following scheme: T a D (23) (12) T T >t to: (re3) (232) 2 a (t3) Proof: This theorem is clear from the manner in which dual and transpose have been defined. | Corollary 5.2.2. If m= wD # wt there exists a third plane iD # gq, wt # aT. Proof: If wen then w= en. If wl = rr, then ho, 5.3. Collineations. Theorem 5.3.1. The collineation group of nt has the same order as the collineation group of 17. Proof: Refer to Theorem 3.4.4. The groups of translations and shears are ismorphic. The autotopism groups are also isomorphic, by Theorem 4.2.3, since [F,c,H] x A= A [¥F,H,G] x al@3) = q(23), (5.1) Therefore, the orders of the collineation groups are the same. The interrelations of autotopisms, translations, and shears are, however, different as equations (3.16) show. | 5.4. Examples. The concept of transpose can be generalized to apply to Veblen-Wedderburn systems, but that will be omitted here. The transpose of a Desarguesian plane is Desarguesian, for all matrices of left multiplication are powers of the same matrix, and this property remains under transposition. The semifield planes of order 32 have been completely tabulated by R. J. Walker [4]. Besides the Desarguesian plane, there are five others. These calculations were independently checked by the present author and some interesting relationships were found. Plane P(1) has the 3-cube representation 10000 01000 00100 00010 00001 01000 00100 900010 090001 10100 00100 00010 01001 10100 oO0101 . (5.2) 00010 00001 11010 11110 10111 Oo000l 10010 11011 10000 01110 Wy No way to construct this plane, except by trial and error, is known. The solution given of all the representatives of the plane has, in some sense, a relatively high degree of symmetry. This plane was discovered in December, 1961, by a program written for a Burroughs 220 digital computer. There are no autotopisms except the identity; hence, the col- lineation group of P(1) consists entirely of translations and shears. This is the only known semifield plane (except for its dual) with this property. A further consequence is that there are 31° distinct semifields, isotopic but not isomorphic (see equation (3.13)). Plane P(2) is the dual and.also the transpose of P(1), and it has the same properties. Plane P(3) is constructed by the method of Section VIII. It has the following 3-cube representation: 10000 01000 00100 00010 o00d01 01000 00100 00010 O0001 11000 00100 00010 11001 01010 0oO111 (5.3) 00010 00001 01010 11012 11100 00001 11000 00111 11100 10111 This particular representative has five automorphisms and no other autotopisms; there are 192 other distinct systems isotopic to this one, and none of these have any automorphisms. This agrees with formila 3.13: 31° = (5/5) + 192+(5/1). 42. Plane P(3) is self-dual since its semifield is commtative, but it is not self-“transpose. Therefore, (Corollary 5.2.2) there are two other planes, P(4) = p(3)7, and P(5) = P(4)>, These planes also contain one system with automorphisms and 192 other systems — without. It has been shown, by exhaustive enumeration on a computer, that these five planes constitute the entire set of proper semifield planes of order 32. 43, VI. SOME KNOWN SEMIFIELDS, In this section, we discuss many of the known semifields and consider what the possible orders of semifields are. Throughout this section the letter p will denote a prime number. 6.1. Orders which are excluded. We have seen that semifields must have p" elements, For each p and n there is a unique field with p"™ elements, What can be said about proper semifields? If the order is p, a semifield must be the field GF(p). Furthermore, if the order is p*, it is well known that a semifield must be the field GF(p*): Let {1,x} be a basis for the semifield; the multiplication is determined by the definition of x*= ax + b. But the polynomial x? - ax -b has no roots in GF(p), else we would have (x-r)(x-s) = x* -ax - b = 0 contradicting axiom A2. Thus, x? = ax-b is irreducible, and the multiplication is that of GF(p?). If the order is 8, one easily verifies that we have only the field GF(8). This must be true since there is a unique projective plane of order 8, but we can verify this particular case directly: Let {1,x,y} be a basis, and let L be the matrix A2** for left multiplication by x. If the characteristic equation of L is Ay, AS + ah? + bA +c = 0, L satisfies this equation, so we find that x(x?) + ax? + bx + ¢ = 0. This polynomial must have no linear factors; hence, it is irreducible and takes one of the two forms x(x?)+x+l = 0 or x(x2)+x2+l1 = 0. Replacing x by xtl, if necessary, we can assume x(x2) + x +1 = 0. In particular, (1,x,x2} is a basis. The remainder of the proof considers various nontrivial pos- sibilities and will only be sketched: it i I. If xx = x*+1, x2x2 = x, then (x2+x)? = 0, x24+x, then (x2+1)(x2+x+1) = 0. II. If x*x = x*+1, x*x? III. If x?x = xtl, x*x2 = 1, then (x?+1)2 = 0. IV. If x¢x = x#1, x*x2 = x, then (x%+x)(x2+x+1l) = 0. Ve. If x@x = xt, x2x2 = x241, then (x2+1)(x2+x) = 0 VI. If x@x = x+1, x2x2 = x2+x, then we have the field GF(8). We have proved the following theorem: Theorem 6.1. A proper semifield has order p", where n> 3 and po>ié. | We will show in Section VIII that these easily obtained, necessary conditions on the order are actually sufficient. 4S, 6.2. Semifields of order 16. The semifields of order 16 have been tabulated in [5], and these have been independently checked on a computer. There are 23 nonisomorphic, proper semifields of order 16. These are all isotopic to either system V or to system W of Section 2.2; consequently, two projective planes are formed. The first plane, consisting of those semifields isotopic to V, contains 18 distinct semifields. There is one (namely V) with 6 automorphisms, another with 3 automorphisms, 8 with 2 automorphisms, and 8 with only the identity automorphism. Hence, there must be 18 autotopisms in agreement with formula (3.13): 2 8 8 3°35 + 1s (6.1) 1 15% = 18( + 6 The second plane has only five distinct semifields. One of these has 4 automorphisms, one (W) has 3, and the other three have 2 automorphisms, Thus, by formula 3.13, there are 108 autotopisms, and 152 = 108(= + = + 3), (6.2) ike Ww ike in) Since the number of autotopisms is different, we can conclude that each plane is self-dual and self-transpose. 6.3. Early work of Dickson. The study of semifields apparently was originated by L. E. Dickson in 1905 [6]. In two early papers on the subject, he considered the construction of all possible semifields of order p?, and of all possible commutative semifields of order p', where p is odd. hé, Perhaps the simplest way to construct proper semifields is analogous to our construction of V and W in Section II; we start with a field GF(p™) and construct a semifield of order p*™, having elements a+ Ab; a,b « GF(p™). One must merely define (a + Ad)(c + Ad) = fla,b,c,d) + Agla,b,c,d) | (6.3) where f,g are linear in all four variables, and where f(a,b,c,d) = g(a,b,c,d) = 0 implies a=b=0 or c® d=0, There are many, many ways to do this, seemingly unrelated; a quite general class of these constructions is investigated in Section VII. Dickson [7] gave a particularly simple construction of a commutative semifield of this type, for p odd: (a + Ad) (¢ + Ad) = (ac + O°aTF) + A(ad+de) (6.4) where o is an automorphism and where f is a nonsquare of the field GF(p™). The condition that f must be a nonsquare element is clear, for if f= a2 choose b such that b° = at). Then (1 + Ab) (1 - Ab) = 0. It is easy to verify that the condition is also sufficient (see Section VII). The complex numbers are a particular case of the system (6.4), although f being a nonsquare is not always sufficient in the infinite case. The system (6.4) is associative if and only if o =I. 47. 6.4, Twisted fields. The following construction is due to A. Aw Albert [2,8,9]. Define a new multiplication on the elements of GF(p™) by xoy = xyt -c xl, (6.5) where g=p™, 1 <m<n, and where ¢ # at} for ae GF(p"). Then we obtain a pre-semifiela, since x2 + y4 = (x + y)%, and since xeoy=0 implies x=0 or y=0 or c= (y/x)@ I, Now pass to a semifield by using (4.19); the result is called a twisted field. The construction can be carried out only if c exists subject to the required conditions. This will be the case when (q-1,p"-1) > 1, iee., when q=1 and p®-1 have a common factor, since miltiplication is cyclic of order p®-l. But if p is odd, there is always the common factor p-1; if p=2 we have (2™-1 ,2°-1) = p(myr)_y so we need (m,n) > 1. If n= mk, where k > 2, it has been shown that the semifield constructed is nonassociative. Twisted fields exist for nearly all orders not excluded by Theorem 6.1.1, The missing orders are 2*, and 2°, where p is a prime greater than 3. 6.5. Sandler's construction. An interesting class of semifields , 2 has been constructed by R. Sandler [10]. There are p™ elements in these semifields, where m is greater than l. 48, For Sandler's construction, let q = p™; the elements of S are eee Ti | m ay + Aa, + + aij? 94 & GF(q"). (6.6) Multiplication is defined as follows: Oddy) = atxty, osicm ogix<e B= 5, (6.7) with the convention that »* denotes left powers of A , i.e., yet = ak, If 8 4s chosen such that it satisfies no polynomial of degree less than m over GF(q), this gives a semifield. For example, let us construct such a system S of order 2%, The elements are a+ Ab + AZ, a,b,c « GF(8). Multiplication is defined by the rule (a + Ab + A*c)(d + Ae + APF) = ad + late + Male + Sv'f + Abd + A2d2e + §c2e + ySc4f + ded. (6.8) Note the similarity between this and the definition of the system W in Section 2.2. This can be written in matrix form, since S is a right vector space over GF(8); the matrix of left mltiplication by (a+ dAb + 2c) ~=is then a 6c2 éb+ L = b a2 éc*l, (6.9) hg. The determinant of L is a’ + 6b? + 6%c7? - S(a*bte + ab2ct + aye?) =r+sé+t6%, Since v7 =r, st =s, t* =+t this is a poly- nomial of degree 2 or less over GF(2), and cannot be zero by hypothesis unless all of the coefficients vanish; but this would. require. a =beF=cr0. 50. VII. WEAK NUCLEI AND QUADRATIC EXTENSIONS, The concept of nucleus (Section 2.6) is generalized here, and this leads to an important class of semifields of the form (6.3). Tel. Definition of weak nucleus. Let F be a field contained in a semifield S. We say F is a weak nucleus for S if (ab)e = a(bc) whenever any two of a,b,c are in F. S will be a vector space over F, but left and right miltiplication will not in general be linear transformations over F, An example of such a system is the semifield V of Section 2.2. Here F is contained in neither the left, middle, or right nucleus of V, There are, on the other hand, semifields of order 16 whose left nucleus is not a weak nucleus. In other words, the statement, "“F is a weak nucleus" does not imply and is not implied by the statement "F is a left nucleus.” However, if F is contained in any two of the nuclei N Na or N F must necessarily be a R? r? weak nucleus. 7.2. Quadratic extensions. Theorem 7.2.1. Let F be a weak nucleus for 5, and let 5 have dimension 2 over F. Then the elements of S have the form a+ db, a,b ce F3 (7.2) 51. and A can be chosen such that ad = dad (7.2) for all ae F, where o is an automorphism of F. Proof: Let x be a primitive element of the field F,. and let xA = a+ dd. (7.3) Case I. b= x. Then we can prove by induction that x™) = nax®! + ax, (7.4) i Rn for x8tly, = (xx), = x(x\) = x(max®7! + Ax®) = nax™ + x( ax?) (n+l)ax® + Ax®*), nax? + (x\)x® = nax™ + (a + Ax)x@ Notice that the associative law was applied here only when permis- sible. If F has p™ elements, we can put n= p™, and since pa =0 and xP" =x, we have by (7.4) Case II. b#x. Then we may replace A by A!‘ = d+ a(b-x)7}, and we find xA' = A'b. We easily prove by induction that x")," = A"b", Since x is a primitive element, b is a power of x, say x°%. This shows that yA't = Aty® for all nonzero y, and in fact (yz)A? = A'(y%2%), . (7.6) 52. for all nonzero y,az since y and 2 are powers of x. Further-~ more, (y + z)At = At(y? + 2°), (7.7) by the distributive laws. These equations hold trivially if. y cr z is zero; therefore, we have shown that (yz)° = y°2%, (y+z)° = y? + 2°, This means go is an automorphism, i.e., o = p* for some k. Jf {-3. Construction of semifields. The rule for multiplication in a quadratic extension of a weak nucleus must be (a + rb) (c + Ad) = ac + A(a%Zd + be) + (Ab)(AAd), (7.8) according to the preceding theorem. But the fact that F isa weak nucleus tells us nothing about what the product ()b)(,d) must be; such products never occur unless two elements not in’ F ara multiplied together. This product can be defined as f(b,da) + Ag(b,d) where f and g are bilinear functions, as long as no zero divisors are introduced. Although many choices might be made for f and g, we will make the assumption here that they have a certain simple form. Let (Ab) (Aa) = vtabe +A(pVaSg) (7.9) where f,g are now elements of F, and where a4,8,y,5 are automorphisms of F. Under what conditions can we conclude that no zero divisors have been introduced? 53. Suppose (atAb) #0, (c + Ad) #0, but (a+ Ab)(c + Ad) = 0. Then, in the first place, we have ac + p"a®r = 0; this implies there is an element x #0 in F such that i a=xd®> 0 ¢ = «x7! b%r, (7.10) The other condition is that ald + be + bYddg O, ie., xIqgBOF] 2 patlygwly 4+ bYa5g = QO (7.11) Case I. g=0. If g=0, we have x%q80t] = potlynly, Here b and d must be nonzero, and f = x°%t}q80tlp-1-9, there- fore, if p is an odd prime, o+tl, Botl, and -l-a are even and we could conclude that f is a square; so choose f tobe a nonsquare. Other a@ priori choices of a,f8 and f ean be made, in which f is a square, as we will see in our discussion of Case II. . For particular fields, special choices may be possible. Case II. g #0. In this case, we will try to choose a,8,y,6 so that (7.11) is a nonzero constant times a polynomial in a single variable. This can be done in essentially one way by writing it in the form x7 pot) ( d+], -1-ag8ot] + xbV~7 1-45, -f) = 0, and we require that xt prleagBorl = (xpY-1-ag6)ot1, (7.12) 5h, Theorem 7.3.1. p* + p? =p + pt (mod p™1) if_and only if aze,bdzd or azd,b=zc (mod m). Proof: Since p™ =1 (mod p™1), we may assume O < a,b,c,d <m., Then we have 2 < p® + p? < apt! < p™, ana C + pd < opal < p™, so we may conclude that nm a to p® + pP = pe + pd, We may now divide by p if necessary, until one of a,b,c,d is zero, say a. Then it is clear that either ec or d must be zero, and the theorem follows. | Now suppose F = GF(p™), and let t= o07!. Then (7.12) can be written Bo +1 = 60 + 6 (mod p™1) (7.13) yo+t y = ao +o (mod p™1)- By Theorem 7.3.1, these two congruences have four solutions: (8,6) = (t#,1) or (1,1) 3 (a,y) = (t,1) or (0,0). (714) Using any of these four solutions, we can conclude that there are no zero divisors if and only if the polynomial ytl+ey-f = 0 (7.15) has no solutions in F, De There are in general many such polynomials, as long as F is not a prime field. For example, if p is odd, we may take g = 0, o =p and f any element not of the form yPt!, If p= 2, we can take g=1 and og =2, and f any element not of the form y? + y. (Such an element must exist, since 03 +0 =19 + 1.) 7.4. The four types. We obtain many proper semifields, which are quadratic over a weak nucleus, by the construction in the previous section. Those of Case I have the multiplication rule (a + xb)(c + ad) = (ac + DAFF) + A(a%a + be) (7.16) where p is odd and f is a nonsquare; a,8, and o are arbi- trary automorphisms, but not all the identity. Dickson's construc= tion (see Section 6.3) is a special case, indeed the only commutative semifield of this type. Other semifields, those of Case II, are constructed by finding an automorphism o #1 and elements f,geF such that yotl + gy -f # 0, for ye F. Then four types of semifields are produced, all for the same o,f,¢g and for 1 07}; (a + Ab) (c + Ad) = 2 I. (ac + boat f) + A(be + ald + DIdTg) Il. (ac + b9d £) + Al(be + aMa + b°%d g) (7.17) III, (ac + btTat’f) + arlde + aa + dD ata) IV. (ac+btd ff) + XA(be + aa + bd a g) 56. For example, the system. V of Section 2.2 is of type I. The system W of that section is actually of all four types. The systems of type II were discovered by Hughes and Kleinfeld [11]. Autotopisms of systems of type II are discussed in [10,12,13]. Theorem 7.4.1. Let S be a proper semifield, quadratic over a ‘finite field F. Let N,N, be the nuclei of S. Then (a) F nN, if and only if S is of type IT. “ s w (vo) FHN Q Nin if and only if S is of type III. i be u (c) F , = N, ifand mly if S is of type IV. Proof; The "if" parts of this theorem can be obtained by direct calculation, and will be omitted. The "only if" parts follow since, first, F is a weak nucleus if it equals any two of the nuclei, so Theorem 7.2.1 holds. We need then merely calculate (xb}(a) under the assumptions (a), (0), (¢) given. Let A2 =f + Ag. Then (a) (Xb) (Ad) = ((Ab)A)A = (A(DA))Aa = (ACABT) Ja = AC (Ado) a) u it A(A(b%d)) = A2(D%a) = Daf + ADTdg, (bTx)(aTA) = BT(A(aTA)) = BT((AaT)A) = BT((aT?A)a) (b) (Ab) (Ad) 2 2 pt (at y2) = (pta™ )a2 = ptat (f + dg) It 2 pat f + dAbalg, (bTA)(ad) = bBT(ACAd)) = b*()24) (c) (ab) (ad) pt((f + Ag)a) = vtaf + Xbag. aT. The proof is completed by observing that either of the three types implies the existence of the polynomial (7.15) which has no solutions in F. | No characterisation of type I is known. Corollary 7.4.2. Under the hypotheses of Theorem 7.4.1, F= Ny = Nin = Ne if and only if S is of all four types I, Ti, III, IV. where o* =1 and g = 0. Proof: By the theorem, we must have S of types II, ITI, IV at least. These types are the same if and only if o2 =] and ¢g=0Q, since o #1. J This corollary and part (a) of Theorem 7.4.1 were discovered by Hughes and Kleinfeld in their original paper [11]. 58. VIII. BINARY SEMIFIELDS. Constructions given in this section fill the gap in the existence theorems for semifields and non-Desarguesian planes of all possible orders p"™. Proper semifields of all orders 2", where n#¥3 and n is not a power of 2, are constructed. 8.1. The binary pre-semifields. Let K = GF(2™) where n is odd, n>1; let K, be the subfiela GF(2™), Considering K as a vector space over Ko» let f be any nonzero linear functional from K to Ky? f(xa + yb) = xf(a) + yf(b), (8.1) for all a,beK, andall x,ye Kee Define a new multiplication in K as follows: a®b = ab + [f(a)d + f(d)al?. (8.2) This product clearly is linear in both variables, so it satisfies both distributive laws. We will show that there are no zero divisors; Suppose aco b= 0, and a,b #0; then let x = ab-!, This implies x + f(a)? + £f(b)2x2 = 0. 296 We have a quadratic equation with coefficients in K3 but since the degree of K/K, is odd, this equation must be reducible. There- fore, Xe Ko But then a= xb implies aob* ab + [f(xb)b + f(b)xb]* = ab # 0. 8.2. The binary semifields. A semifield can be obtained from this pre-semifield in any of the standard ways described in Section 4.5; let us call any such semifield a binary semifield. We will give a particular construction in detail here. It is necessary to do this, to make sure the resulting semifield is not a field. Let {1,x,x?,...,x""!} be a basis of K over K,; set £(1) = £(x) = eee = f(x8°?) = 0, £(x*1) = 1, (8.3) With this definition, we find that, for ye K., loy=y,10 yx = yx, ..-, 10 yx = yx, Lo yxt@t = yxtwl + y?; and therefore lo({loa)=a, all ack. (8.4) Define the multiplication , by the rule axb={loa)o (lob). ~ (8.5) Then 1 is an identity for a*b, and we have a semifield (S,+,*). 60. Notice that we have now defined three different multiplications on the elements of K: ab, ao°b, and as#b. It is important to keep this distinction in mind, since all three multiplications are used simultaneously in many proofs. The powers of an element a’, a, etc., will always refer to the multiplication of the field. Theorem 8.2.1. S is a proper semifield of order 2™, if n is odd, and if mn > 3. Proof: If n> 3, let k = (n-1)/2; then k > 1, and x x (x® y xX) = xy xl ¢ x = (x y xX) a x, Thus, multiplication is not associative in this case. If n= 3, let y be an element of K,; we have (x » x) * yx = x? « yx = (x2 + L)oyx = yx(x2+1) + y2x?, x x (x * yx) = x « yx? = x o (yx2+y2) = (yxtty2)x + y?x?, Thus, multiplication is still not associative, unless y* = y. We can always choose y # y* unless K = GF(8), in which case we have shown before that no proper semifield exists. | Corollary 8.2.2. (Converse of Theorem 6.1) Proper semifields exist for all orders p", where n> 3 and p*>16. | Corollary 8.2.3. Non-Desarguesian projective planes exist for all orders p™, provided n> 2 and p™> 9. Proof: Hall systems (Veblen-Wedderburn systems) exist for the remaining orders p*, p odd [| 61. 8.3. Isotopes. The definition in 81 involves an arbitrary linear functional f. In this section, however, we show that all of the binary semifields are isotopic, so only one plane is determined. Theorem 8.3.1. if f,g@ are nonzero linear functionals from K to K Ko» there exists an element zeK such that f(az) = g(a), for all a in K. Proof; A simple counting argument applies. Let (x, sXpye00%,)} be a basis of K over Ky. A linear functional is completely specified by giving f(x,;), 1< isn, and these choices are independent; hence there are 2°" —~j nonzero linear functionals. Suppose f is a nonzero linear functional; then f(xz) is also a nonzero linear functional, if z2#0. There are 22M. ] choices for 2, so we need only show that no two of these give the same function. But if f(az,) = f(az,,) for all a, we have f(a(z yp) = 0 for all a, hence Zz, - 2, = o. | Theorem 8.3.2. All binary semifields, for given K and K,, are isotopic. Proof: We need only show that any two of the pre~semifields are isotopic. Suppose we have ul aob=ab + [f(a)b + f(b)al? ab + [g(a)b + g(b)a)?. © a " 62. By Theorem 6.3.1, we find ze K such that f(az) = g(a) for all a Then we have az © bz = abz? + [g(a)bz + g(b)az}]* = (a * b)z2. | 8.4. Automorphisms. The binary semifield exhibited for the plane P(3) in Section 5.4 has 5 automorphisms. This property holds in general: we will show there is a binary semifield with n auto~ morphisms. In this section we let q = 2™. Theorem 8.4.1. Let f be such that f{a}) = x whenever a=xtb+bt, xek be K. (8.6) oo? Then f is a linear functional from K to Koe Proof: First we show that f£ is well-defined. Suppose x+b+bts=y+e+c% for x#y; then (btc)2 = (b+c) + xty, i.e., at=a+z, for some ac K,2e€K,. Applying this rule again, we find at = ol 424s alte =a. 1 n+1 ae But ad = al, hence "i : : n+ Since n is odd, we find at z= 0. Thus f is well-defined. Furthermore, £(xtb+b% + yt+etc?) = £(xty + (bte) + (b+c)4) = x + y f(x+tb+b2) + f(y+ete?). f(y(x+b+b%)) = flyx + yb + (yb)%) = yx = yf(xt+b+b2). This shows that f is a linear functional. | 63. Theorem 8.4.2. Let f be defined as in Theorem 8.4.1, and let the preduct aob be defined by equation (8.2). The mltiplication +« defined by (loa) * (lob) =aobdb (8.7) has the automorphism a-+>a‘%, hence there are at least n auto- morphisms of the binary semifield. Proof: We show first that a-+ a2 is an automorphism of the circle product aob. (By at we mean the multiplication of the fiela K.) (a ° b)2 = (ab + [f(a)b + F(b)a]2)% u alb2 + [£(a)b4 + f(b) at]? since f(a) = f(a2) by definition. The automorphism carries over to the star product, since © il (10a?) 4 (lop?) (190a%) * (14%0b2%) = (loa) * (lob). | 8.5. Autotopisms., Let us investigate the autotopism group of a binary semifield; it suffices to consider autotopisms of the pre- semifields. The following theorem is an important step in this direction: 6h, Theorem 8.5.1. If [F,G,H] is an autotopism of a binary pre- It semifield, we have F=Gz, for z2#0 in Ky; i.e., i aF z(aG), for all ae K. (8.8) Proof: We have aF 9 bG = (aob)H = (boa)H = bF © aG, for all a,b. Therefore (aP)(b6) + [f(aF)bG + f(bG)aF]? = (aG)(bF) + [f(aG)vF + f(bF)aG]?. (8.9) Let V, = {a | f(aF) = 0}, Vo = {a | f(aG) = 0}. Vv, and V, are vector spaces of dimension m(n-1) over -GF(2), since the kernel of f: K-+K, must have dimension n-l over K,. Let V3 = Wa Vos Then dim V, = dim V, + dim V, - dim V, u Vo 2 2m(n-1)-mn = m(n~2) 2 2. Therefore V2 contains at least two nonzero elements. We apply formula (8.9), to find, (aF)(bG) = (aG)(bF), if f(aF) = f(aG) = f(bF) = f(bG) = 0; Leese, af = bE . z, for all a,b EV - {OQ}. © (8,10) aG bG Now, let ae Vz ~ {O} and let b be arbitrary; we have 2(aG)(bG) + [zf(bG)(aG)]? = (ac)(or) + [£(oF)(aG)]?, Leen, z(bG) + bF = (aG)[zf(bG) + f(oF)]*. (8.11) 65.6 Let a run through two distinct elements 85 80 of V3 ~ {O}; the left hand side stays constant, hence we find (ajG ~ a,G)[zf(bG) + f(vF)]? = 0. But G is 1-1, s0 a,G ~ anG #0. This shows us that 2f(bG) + f(bF) = 0, all ob. In particular, if b ¢ Vo. we find ze Kk. | Finally, equation (8.11) becomes z(bG) + bF = O, all b, and this is precisely equation (8.8). | Remarks: Put a= b, we find (ao a)H = (a*)H = 2(ac)?, so H is given in terms of G,. Other relationships can now be obtained, using methods similar to those of the preceding theoren, but this investigation is incomplete at the present time. References a) [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] pp. 346-420, M. Hall, Jr. The Theory of Groups. New York, Macmillan, 1959. A. A. Albert, "Finite division algebras and finite planes," Glinter Pickert. Projektiv Woenen. Berlin, Springer, 1955. 343 pp. . R. J. Walker, "Determination of division algebras with 32 elements," Proc. of Sym. Exp. Arith., to appear. Erwin Kleinfeld, "Techniques for enumerating Veblen-Wedderburn systems," J. ACM 7 (1960), pp. 330-337. L. E. Dickson, “Linear algebras in which division is always uniquely possible," Trans. AMS 7(1906), pp. 370-390, 514-527. L. E. Dickson, "Linear algebras with associativity not assumed," Duke Math. J. 1(1935), pp. 113-125. A. A. Albert, “On non-associative division algebras," Trans. AMS 72 (1952), pp. 296-309. A. A. Albert, "Finite non-commutative division algebras," Proc. AMS 9(1958), pp. 928-932. Reuben Sandler, "Autotopism groups of some finite non~associative algebras," Amer. J, Math. 84 (1962), pp. 239-264. D. R. Hughes and Erwin Kleinfeld, "Semi-nuclear extensions of Galois fields," Amer. J. Math. 82 (1960), pp. 389-392. D. R. Hughes, “Collineation groups of non-Desarguesian planes II," Amer. J. Math., 82(1960), pp. 113-119. A. As Albert, "On the collineation groups of certain non—Desar- guesian planes," Portugaliae Mathematica, 18(1959), pp. 207-224. D. E. Knuth, “Non-Desarguesian planes of order 22™*!," (abstract) AMS Notices 9 (June 1962), p. 218.