Periodic Functions over Fields of Characteristic p Citation Swift, Jonathan Dean (1947) Periodic Functions over Fields of Characteristic p. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/n2pz-dd14. https://resolver.caltech.edu/CaltechTHESIS:03312025-162559488 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Bell, Eric Temple (advisor) Ward, Morgan (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1947 Record Number: CaltechTHESIS:03312025-162559488 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03312025-162559488 DOI: 10.7907/n2pz-dd14 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17114 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 31 Mar 2025 19:51 Last Modified: 31 Mar 2025 20:04 Thesis Files PDF - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

Periodic Functions over Fields of Characteristic p Thesis by Jonathan Dean Swift In Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy, California Institute of Technology 1947 Acknowledgement I wish to express my sincere appreciation of the help and guidance given me by the staff of the Department of Mathematics of the California Institute of Technology. In particular, I wish to thank Professors Bell and Ward for their interest in, and encouragement of my work. The suggestions and assistance given by Professor Ward in the preparation of this thesis have been invaluable. Sunnnary A function from a field of characteristic pinto the same field is called periodic with period modulus M, if f (x+ ),l-) = f (x), where f is any element of M. For a Galois Field, GF(pt), there exist non-constant functions, periodic with any proper sub-modulus of the field. For any field of characteristic p, and any set, a , ,aL,•••, a~_,,a~, of elements, linearly independent with respect to the prime sub-field TT, n <. t, the functions: fca,;~) = )C~-ar-•x t(a,,a1.;x) = f(f<a,;a. 1 ) ; f ca,;x.)) f (a, ,a2., ... ,an; x) = f (fca,, ... , a'\'\-,; a"'); f<a,, ... ,a"_' \ ')( )) form a set of per~odic functions with moduli: (a 1 ),(a,.,a 1 ), .. ,. (a,, ..• ,an)• These functions are additive, linear with respect to 1T, and symmetric in the a\. Any function periodic with the modulus Mn in a GF(pt) may be represented by: where f(Ml'\;x) ::. f(a,, ..• ,a\'\;x), Mr,.= (a,, ... ,a"') and the bi are elements of the GF(pt). The latter part of the thesis is devoted to the consideration of other possible bases for representation of all periodic functions over Galois fields. A definition of an orthogonal representation is given and sets satisfying the d e finition are exhibit ed for certain special cases, in particular fort even and n = t - 1. More generally, it is shown that, ir the GF(pt) is imbedded in the splitting field of a certain equation, a set of polynomials exist such that their values for elements of the GF( p t ) are in the latt er fie ld and are orthogonal with respect to that field, although the polynomials require, for their simplest expression in terms of the fi(M;x), coefficients fr om the s uperfield. TABLE OF CONTENTS l. Fields of Characteristic p. Basic Concepts .. . . . . . . .. ... . l 2. Periodic Functions • . • • • . • . • • . • • • . 2 3. Singly Periodic Functions • • . • . • . . . . . 5 4. The Doubly Periodic Function . . . • • . . • . 10 5. The Basic General Periodic Function . . . . . 14 6. The G neral Representation Theorem • . . . . . 17 7. Other Basic Sets of Functions . . . • . • . . 20 8. Orthogonal Systems of Functions • . . • . . . 24 and Definitions • • 8 9. 10. . . . . . . . . . . . . Unsolved and Unfinished Problems . . . . . . . Miscellaneous Results Appendix 39 . . . . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . a. Table of finite fields b. The theory applied to the fields, GF(32 ) and GF{3 3 ) c. 36 • 41 • • • • • • • • • • • • • • • . 46 The number of periodic functions for . . . . . . . • • . 48 Bibliography . . . . . . . . . . • • • . . . . . 49 certain fields • • • (1) 1. Fields of Characteristic E· Basic Concepts and Definitions. All fields contain one and only one prime field, TI. This prime field is either isomorphic to a residue class modulo a prime integer p, or to the field of all rational numbers. In the first case, the field is said to have charac- teristic p; in the second, characteristic O. If a field of characteristic p has a finite number of elements, it is called a Galois field. All such fields of given order are abstractly identical and have pt elements where t i s a positive integer; t i s the "index" of the field, GF(pt).( 3 )-i:The majority of the the orems to be deduced will concern Galois fields. Theorems applicable to all cornmutative fields of characteristic p will be specifically noted. The following equations, valid in any field of characteristic p, will be used in the sequel without further comment: Here the at are any elements of the field. The GF(pt) may be constructed by selecting a polynomial, irreducible modulo p, of degree t, and considering the pt * Numbers in parentheses refer to the corresponding entry in the bibliography. (2) residues modulo the polynomial and p. A tabulation of all quadratic and cubic fields, (t -=-2,3) such that pt < 100, is given in an appendix. The additive group of the GF(pt) is of type (p,p, ... p). Thus the subgroups are composed of pk elements, k ~ t, and are generated from a basis, (a,, ... ,ak), of any k elements linearly independent with respect to TT .( 2 ) The multiplicative group of the field is cyclic. If pt= q, all members of the GF(pt) satisfy the equation: x4 - x = .Q .. Each element of the field possesses one and only one pth root, and, thus, one and only one pk th root. Further, if d = (m,q - 1), exactly (q - 1)/d elements are mth powers (i.e. have mth roots) in the field.Cl) 2. Periodic Functions. In this section the notion of periodic functions over a field of characteristic p will be defined and some general properties of these functions will be discussed. The dis- cussion will, of course, be confined completely to algebraic properties. Although a formal derivative exists, the lack of a valuation makes an analytic discussion impossible. In general, very little attention has been paid to the existence and algebraic properties of periodic functions in fields outdise of the customary real and complex number domains, although functions analogous to the common periodic functions, (3) but not necessarily periodic in the new domain, have been studied.(27,28,29) A large amount of work has been done, particularly by Garlitz, on the properties of special classes of polynomials in a Galois field.( 4 )-( 25 ) inc. In general, this work has dealt with the polynomial domain itself, using the literal elements as indeterminates. In the present work, a special class of polynomials is considered not only for its own properties as polynomials, but also from the standpoint of the values attained when the variables are replaced by elements of the field. For Galois fields the limitation of the functions considered to polynomials is no real restriction since all functions defined for all members of the field with values in the same field may be represented by polynomials with coefficients in the GF(pt). Definition 2.1: A funct ion, f(x), defined over a field of characteristic p with values in the same field, is called periodic if an element a of the.field exists such that f(x+a) = f(x). The element 'a' is called a period of the function. Definition 2.2: A function f(x), defined as above, is k-tuply periodic, with periods a ,,a~,·••a~, if f(x+m 1a 1 + m1 az + .•• +mkak) =-f(x), where them~ are arbitrary (4) members of the prime sub-field, independent with respect to TI, and the at are linearly Tr. The definitions have been phrased in the above manne r to preserve the analogy with the ordinary case. However, it is clear that, if a is a period, the elements 2a, 3a, ... , (p - l)a are also periods. That is, instead of a single elementary period, there exists a period modulus M, where M is the modulus generated by a,, ... ,ak• Since the field is unordered, there is no particular period singled out from the modulus as the period of f(x). Definition 2.3: We thus have: The largest modulus generated by the periods of a periodic function is designated the period modulus, M, of the function. For any }A in M, f(X+J,1) = f(x), for every x of the GF, and for any JJ- not in M, f(x+ fl) f. f(x) for at l~ast one x of the GF. If the modulus consists of pk elements, the function is said to be k-tuply periodic. Any linearly independent set of k elements of M is. called a (complete) set of periods. Theorem 2.1: GF' (p t) A non-constant function, f(x), over a has at most t - 1 independent periods. Proof: If the function had t periods, M would contain pt elements, and would thus include every element of the GF(pt). Since f(x + fl) = f (x) , Theorem 2.2: f- ~ M, f(x) would be constant. Given any pr oper submodulus, M., of the ( 5) GF(pt), there exists a (at least one) function, f{x), such that Mi s the period modulus of f(x). Proof: We construct such a function as follows: the order of M be pk. Let Choose a complete set of periods, a 1 , • • • ,ak., and choose t - k other elements , ak+i, • •• ,at, such that the set, a,, ... .,at, form a linearly independent set., that is., each x of the GF(pt) may be uniquely represenx = ted as L. m. a , . i. l l We then define f(x) by: t = .,:k+I [ ml a i. • f ( x) This is a function of the required type for., clearly, the addition of an element of M to the argument does not change the corresponding value of the function., while for any modulus M') M, the value is changed by addition of an element of M' which is not in M. The appendix contains a description of a notation which facilitates the choices described above, and illustrations of functions using this notation. It is clear that, in general, many other functions of period modulus M exist. For example, a constant could be added to the value of the function for each argument. 3. Singly Periodic Functions. The special theory, developing representations for all periodic functions over a modulus M1 consisting of p elements will now be developed as a basis for the general theory ( 6) covering arbitrary moduli. of M. Let a be a generating element Then: Theorem 3.1: xP - ap-l x is periodic with modulus M over any commutative field of characteristic p. Proof: }A-= ma, m E 1T, then: )'- E: M, i.e. Let (,,c ""y.)'p - ~t,-1 ( )( +f-) = (,c +Ynal - a._,_,(,c + rn~) =- xP - a.P-'x + mpa.r, - mo.I>; mP=m = "I'> - a.~-· t. Corollarz: ,,k x t,k- I - a x is periodic with modulus M. The question arises as to whether a multiple periodicity may be concealed by the use of the above function. That is, is M the actual period modulus? If Mis not the period modulus, there is ab¢ M such that: If k = l, this states that ( ab )~-I = ) or ab E TI, which con- tradicts the assumption, b ¢ M. condition that If k > l, the necessary bpk-1 =- o. pk-I , and b # ma in a GF(p t ) is that (p t-1, :p t ... 1) > p - l. In this case there exists a (7) sub-field of order pk , and here the second type of function may be multiply periodic. This question will be returned to when the general doubly periodic function is obtained; the above function will then appear as a special case . Notation: From this point on, the notation, f( ) wil l be reserved for certain specific types of functions of which the function of Th. 3.1 forms the first example: 1 f ( a ; x ) :: x., - at-- x. . Theorem 3.2: f(a;bx) =. b"'f(~ ;~). Proof: b'\/ - at>-'bx =- bP( ,t- ( t l- x). 1 (Note the analogy with the sine or cosine function of ordinary analysis. I.e. sin bx has a period 1/b of the basic period.) Theorem 3.3: with respect to Proof: f(a;x) is 1) additive, and 2) linear TT. Evident on expansion of f(a;x+y) and f(a;mx). k Theorem !2..!.!= i bl [ f(a; x)J ~:O . L is periodic, with period a, where the bi are arbitrary elements of the field , and where, for convenience of expression now and in the sequel, the convention [ f (a; )()] Proof: 0 = I is adopted. Clear. Theorem 3.5: Any function which is periodic with (8) modulus M J a, over a GF(pt), may be represented in the form: where s . . pt-•. Proof: a period. Let g(x) be a periodic function with a as Select, from the field, elements, a 1 , ••• ,at, such that a, a~,··•,at form a linearly independent set. t The sums, t:= -mi.le. a.~ , where the ml\i.. range over TT, ~c2. form a set of p t-l distinct elements which includes no element of M except O. For these non-zero elements, • .,,._\ X > x.r.,,._, = a"'-' f( a;x ) ..J. t- 0 ; f or, otl1erw1.se, X I> :. a.r ) and ! € TT, a contradiction. The values of the function for the elements of the generated set completely determine the function. We thus have a set of equations: This is a set of s equations to be solved for s values of the bj. The determinant of these equations is clearly. of the Vandermonde type: 0 0 0 [fCa;fml,a.i.)] 5 1 - (9) The value of this determinant is then: TIk f (a; E-mik at> kTTI j [f (a~[ -n, .. 3,.) - f (a· L m.k a.:)] , lJ I. ) l l • This product is clearly non-zero, if: j i: k. f(a;Ln\ja.i.) i:- fca,L1n,ka.i), fca;L mi.jai) =- Lmi/<a;ai) But: L.m .. f(a•a -)::; [lrt. fca · a .) Suppose: Let mq by Th. 3 .3. L) ) 1.k I. , l Lem..'J - n,.k> f ca. ,i a. -)" = 0 • ' - mik = mil : Lmu f Ca; a.i) -=- 0 f ca> L l'\I ai> = 0 and, thus, all ml~ o. that j j k. This contradicts the assumption Therefore the determinant is non-zero, and the solution is possible. We have now exhibited a complete set of functions, 1, f(a;x), f~(a;x)., •.. , f 5 -' (a;x)., which suffice for the linear representation (with respect to the complete field) of any periodic function whose period modulus contains a. The specific set is chosen since it arises naturally from the important function: xP - x = f{l;x); f{a;x) = aPf(l;x/a). The properties of these and other complete sets for representation will be investigated after the development (10) of the theory of representations for the complete period modulus. 4. {The theory of multiply periodic functions.) The Doubly Periodic Function. Before proceeding to the general case, the modulus of order p2 will be discussed. This case will serve as a guide to the higher orders, and is also interesting as the last in which the basic functions may conveniently be displayed in their entirety. Theorem 4.1: Let (a,b) be a basis of the modulus M, of order P2, then: .,1-, pa._, "i. b - a ~ ~-• bti-• ( b"-' .,., ,,., X ------x + a -a:> x bt>-•_ a"-' is periodic with period modulus M. Proof: Denote the function by f{a,b;x). m,n E. TT :f (a, b;x +ma+ nb) = Then, for (11) =f(ab·x) ) ) Remarks: 1) At least formally, the requirement that a and b be independent, is unnecessary. and d =b l>- 1 Let c-:: a.1>-l in the original expression of the function; d~+1 _ c\HI \> the fraction is then: - -- - =- d +d d-c 1>-1 i c +•··+d c l>- + c P • Now, if c = d, each term is cP, and the sum is cP . fore the function is, formally, xl>l_al>i-1> )( 1> There - ==-lfca;x)]p which is clearly periodic with periods a and b : ma. ( 12) 2) It will be noted that the reserved notation was used for the function just established as doubly periodic. The r eas on for this notation will be apparent in the immediate sequel. 3) (Cf. remarks following Th . 3 .1 . ) If a Pa_, - b Pi_, a p-, '/: b p-, , the middle, or xP, term will drop out. final coefficient: - The -... i--1 bp-l Cb f:>-• - -:i, P-·') p-1 Thus the function reduces to the degenerate case: of the singly periodic type considered in t he corollary . Theorem 4.2: f( a ,b;x) is ident i cally equal to f(fca;b)>t(a,,o) ' i . e . t o: (x. P-aY-'xt -{ bl> -atH b>t>-, (){"-ar-, x). Proof: term: If a P-' : b p-, , the theorem is trivial. ( b P - aP-'b)"-' = o, and the forma l r educ tion of f (a, b; x ) to ( r (a;x) ]P has already been demonstrated . Now : The ( x.P -a.~-'x)P -(bP-a_l>- 1 b)l'- 1 {x.l- -a,11-1x) , (13) The func-r; i on, 1'(a.,o;x), as originally written, is clearly ~ymmetric in a and b if p is odd. The symmetry is also immediate when p = 2 since, in this case, the signs may be replaced by + s igns. The symmetry of the second formulation is not directly evident but is established by the identity just proven. Corollary: f(a,b;x) = f(f(a;b ) ;f(a;x)) = f (f(b;a) ;f(b ;x)). The substitution operation used here may be compared to the basic operation used by O. Ore(~l) in connection with his work on polynomials over a finite field. The polynomials here are p-polynomials in the sense of Ore with coefficients in a finite field. For pt-polynomials, (14) Ore defined: F{x) G{x) = F(G{x)). x Here, a double substi~ tution on the two arguments of a single function, f(a;x), is used. 5. The Basic General Periodic Function. Continuing the process by which f(a,b;x) was obtained from f(a;x) or f(b;x), an extended set of functions may be obtained: Notation: t (a,\ X) "' x" - al>-• x 1) fca,.,a2,; ><.) ::: f Cf(a ;a2.>; f ca,;x>) 2) 1 3) f(a,.,aa..,a • ••• • n) 3 ; x) "' fff ca,\a.z ;a. •••• •• •• 3 ); f<a,.,ai.; le)) •••• fca,') ···> al\ ;x) = Hfca ..,... )an·I ,al\); f (an-··-,al\-1 ;>O). - Theorem 5.1: f(al, .. . ,an;x) is independent of the ordering of the ai.; i.e. symmetric in the ai.. Proof: The theorem has already been noted for the case n = 2; assume the theorem proved for all cases~ n - 1, and consider f(a, , ..• ,a~;x) and f(b, , .•. ,bn;x) where the b's are a new ordering of the a's. Now, f(b 1, ••• ,bn;x): Hence, by the induction hypothesis, the b~, i < n, may be rearranged to fall in the same order as the a's. If ai. = bn, the rearrangement may be carried out so that: (15) the original function may, by the same reasoning, be re- a,1-1 .,a i. ,an ;x).. If we designate the two rearranged functions l) and 2): -rf (a,, ... ,a. i-1,a i.+t ') .. ")al\\ a ;,1] ~-· f ( a 1, ••• ,a i.-1, a Lt I , .•. ,ah; X) . 2): [f<a,r•,ai._,,ai+,-,··· )al\_,.,a.i.; x)] t, -[ gt-( ai)- g.--, (a") g a i)]""'[ g\x) - gP·' (a") g (x)]. 2) =[ gP (,c.) _ gti·•(a,) g (,<)] P -[ g"can) _gtt·'ca.l) gca.l\)] i>-• [g" (,c.) - g ►·•caJ g ( x)l Let r :. [gl-·'(aJ-s"·'ca.n)J. Then l) - 2) = g to< x)y-"- s r--•ca.i.) r p-, [g ~(x) - 8 i,-, (ah) g (x)] (16) Thus, since 1) and 2) were (identically) equal to the original functions considered, these functions are also identical, and the theorem is proved . Theorem 5.2: f(a 1 , • • • ,an;x} is additive, and linear with respect to TT. Proof: The theorem has been proved for n : l; suppose it true for n - 1, and consider f(a 1 , • • • ,an;m{x+y)) -m[f Can- .. )a.n_,>an>]t'-• Hca,, ... ,all-• pt)+ f(a. 1 , ... ,an-,;y>] =m\f ~ca,, ... .,an_,;x)-H<a,,... ,ah-•)a0 )]tt-• f(a,,...,an-, ix) ( 17) Theorem E..!.2: f(al, .•. ,an;x) is periodic with the period modulus generated by the ai as basis. Proof: By Th. 5.2: = fca,')···,al'\)x) +[. mi f (a,, ... )al\;a). Now: f ca, ,a.~')· .. ,a >a.") 11 = 0. But, by Th. 5.1, f(a 1 , ••• an;x) is syrnmetric in the at and, thus, any ai may be placed in the final position of the period group. Therefore: f Ca,,a:zy··,ah ; at> = 0 fca,,... )an1 )( t Ll\'\lat) = f(a,') ... ,a t\ )X). and 6. The General Representation Theorem. In the last section, a function, periodic with any desired period moduius, has been exhibited. The further problem of exhibiting all periodic functions with a given period modu lus will now be considered together with the relationship between a function periodic over a modulus, M, and one with an associate modulus, bM, where bis an arbitrary non-zero element of the field. (18) Notation: Let Mn be the modulus generated by a, , ... ,a~. f(a,, •• • ,an;x) Theorem g : = f(M"';x). f(M'h;cx) = ct>" f(c-• M"';x); c is an arbi- trary non-zero element of the field. Proof: it for n - 1. The theorem has been proved for n = l; assume f (a,, ... ,a"'; t x.) Thus, k Theorem 6 .2 : modulus Mn• Proof: Evident. to bi [ f <M,.; x.)t is periodic with (19) f (M"';x) + O, if x ¢ M . Theorem 6 .3: Proof: The theorem has been noted for n = 1 in the proof of Th. 3.5. Assume it for n - 1. previously used, if: By the method f(a , , ..• ,a "' ;b) = O, f ( a 1 , • • • , a 1\ _ , ; b ) = mf ( a 1 , • • • ., a -n _ 1 ; a"' ) . Then: f(e. , , • ., .,,ah- I ;b - ma.,) and b - man f:(a,, ••• ,a"' _1 ) , or b € (a , , ••. ,an). Remarks: 1) -= 0 by Th. 5.2 by induction hyp.; To this point the theorems deduced in the last two sections have l) been valid for any commutative field of characteristic p, and 2) not depended on the independence of the a i • If dependent a i are included, the resulting function will, in general, be a pk power of the basic function for the modulus, but the content of the theorems is unchanged. The following general representation theorem, however, makes use of the elementary fact that a minimal basis exists (components linearly independent with respect to TT), and is valid only over the GF(pt). This theorem is a generalization of Th. 3.5 in that it gives a constructive method for the representation of any function with a given period modulus in terms of the basic f-function for the modulus. Th. 3.5, of course, gives a complete representation (20) theory in the sense that any periodic function over any modulus whatsoever can be represented. The purpose of the next theorem is to provide a representation in terms of functions appropriate to the full modulus. Theorem .§.d_: Any function, g(x) whose period modu- lus has the basis, a,,a 2 , ••• ,a~, over the GF(pt), may be represented by: S-1 gcx) =?; bJfca,,.... la";x>]\ Proof: Select a complete basis for the field: a 1 ,a 1, ••• ,a~,c, ,c~,•••,ct-n• The values of g(x) are completely determined by the values assigned to the pt -~ elements: tnik E. Tf. This yields pt-n equations with exactly the same conditions for linearity and non-vanishing determinants as obtained in Th. 3.5. The theorem then follows in precisely the same manner. 7. Othe1" Basic Sets of F unctions. 1 The functions f(t\;x) used in the previous investigations as the basic functions were chosen for the following reasons: 1) They arise naturally from the important, and 'primitive', periodic function: xP - x. 2) The powers of the f-functions are easily shown (21) to form a complete basis by means of the Vandermonde determinants which arise . 3) The operation of changing the modulus to a more inclusive one results in a simple and interesting operation on the basic functions. However, of course, other bases exist. a set, ¢, "> ••• , ¢s In general, . , S -- pt-l'I. will be a basis over M~ if and only if the determinant of the c.Jl. is not zero. If this condition is met, the ri can be solved for in terms of the ¢j• Then, clearly, any function expressed in the f's can be expressed in the ;,' s . On the other hand, if the determinant is zero, the ¢j cannot form a complete set since some periodic functions, e.g. the fl cannot be expressed in terms of the ¢ ,s. In the development of general sets of ¢'s, the Vandermonde type of determinant is of greate st interest since this type furnishes a guaranteed non-vanishing condi tion and, due to the peculiar behavior of additive op erations in a GF, is essentially the only such type available. The primary question regarding a basis for the repre sentation of periodic functions is: onal?" "Is the basis orthog- That is, is there an analogue of the fundamental (22) properties of a set of orthogonal functions in analysis: J ..L . ..1... .,.,~ ~, = J .. lJ ? • Defin1 tion L.1: A set of functions, tpi.(."f.J, periodic with a modulus Mn, will be called a completely orthogonal set if: 1) The set forms a complete basis for the representa- tion of all periodic functions with Mn as modulus, and [ = 0 if i :/;- j I and L ~t ~ o) GF Gr +;. ~j the SUlll being taken over all elements of the GF(pt). 2) Theorem 7.1: Proof: No completely orthogonal set exists. Consider any member of the set, ¢L. is periodic with Mn as modulus, +f is also. [~!CK) : Gf Since 'A But then 0, for each value which ¢f takes on is assumed pn times by reason of the periodicity. This result, however , remains analogous to the results with the usual periodic functions. The orthogonality principle for sin nx, for example, applies solely to integrals (sUllls) taken over the range of a period, rather than over the whole field. In this case, a 'period' will correspond to a maximal set for which the function takes on no repeated values which are induced by the periodicity. That is, a set of elements bL such that no two elements, bt, bj, are 'congruent mod M'II' ; bl - bj € M implies i =- j. (23) The simplest such set is a complement of Mn,Mn, obtained by writing a complete basis of the field: a,, ~ ... , B.n, b,, ... , bt-n, and considering the modulus, (b,, ..• ,bt_,,). Definition 7.2: A period interval is a set of pt-n elements such that the difference of no two of the elements is coatained in M~. Theorem 7.2: If~(~) is any periodic function with modulus M", and if d ,, ••• , d , t-1'1 is any set forming a period interval, then the sum of the values of ~Cxl over the set of the di, is equal to the sum over Mn = (b,, •• • ,bt_,,} ... b 1 , • • • , b 11t. • ti . Proof: It will be sufficient to prove that ~ Cbl) ... ¢ (di.) where the d's are, if necessary, reordered. Now di."" b" -t ai., where ai is a member of the Mn, since the complete modulus of the field is the direct sum of Mn and M,, . Therefore, .by the periodicity. Furthermore, the set of b's must be exhausted; for, suppose di.= bi.+ ai., and dj :: bi.+ aj, then di. - dj £ M contradicting the assumption. Thus, in the further work., the interval for summation, corresponding to (- TT., TI) for sin rue, will be the set M,,. Definition 7.3: group to M,,.,Mn• A basic interval is a complementary (24) Definition 1..:1,: A set of functions, ¢t(x>, with period modulus M~, is said to be orthogonal if: 1) The set forms a complete basis for representation of periodic functions with modulus M.,,, and 2) the summation being taken over all elements of Mn• This definition, as will be shown in the next section, is non-vacuous. 8. Orthogonal systems, as defined, exist. Orthogonal Systems of Functions . Theorem 8.1: The set: is not an orthogonal set. Proof: If the set consists of more than two functions, select i and j so that i + j = 2k. contradicting the basic condition. Then, if: If there are only two functions, 1 and f (Mn;x), theorem 6 .4 shows that f2 {II\, ;x) is a simple multiple of f{Mn;x) and, thus, the sum of ¢,~i and of ¢ ~ will vanish or not vanish together, which is again a contradiction. Th~~ 8 .!2: For the case t ~ 2, the following set (25) of functions is orthogonal: where we define o0 = 1 to simplify the notation. I.e., ,is_, :: ,,_, f ( a ; ><). To aid the proof of this theorem, two le~.mas will be proved. The first of these is also valid when t > 2. For all i < p - 1, L f\a,'IC) = 0. Lemma l: R Select a sub-set of M, consisting of the Proof: multiples of a single element, say bj, by the elements of TI. Subdivide Minto mutually exclusive classes whose repre- sentatives are bk+ mbj . The sum of ri(a;x) over each class is equal to O, for: p-1 = f)f cbk) + rnf <bj>Ji· = ~t" cbk) + i.f'- cbk)f cbj> L.m 1 i.(i.-t) fi.-Z b f b ~ + ~ ( k) · ( j) L lTI 2. +··· 2 11-1 and [ '11:0 mJ is divisible by p when j < p - 1. For t = 2., [ ft,-•ca;x) :/: 0. Lemma 2: M Proof: If the sum were zero, then, for any function, g(x), with modulus M = (a), ,,_, . i M g(,c) : ~i r, L•O Oli. fl(a;X) = (26) But this is impossible since a function, g(x), periodic with modulus M can be constructed so that the function over M has any assigned values whatsoever. The proof of the original theorem will now be given for p odd. The case 2 2 can be proved in an analogous manner, or simply verified. l) The set is a basis. The determinant is: (2a.>"'~-5) " Tl [crn-n)aP] :/ 0. m>n 0 0 0 For, 3) (~·•),fi <p-1) I l.. (27) The lemmas have shown this for 1 ~ p - 1, but 2) demonstrates that all powers of f(a;x) are reducible to powers ~P - l, and that only those powers which are multiples of p - 1 reduce to 4) fl>-•ca ;1.). Thus, in the expansions of 'Pk ~ 1 , and reduced by the use of 2), only the coefficients of need be considered. In 1 ¢ k, f P·•(a;x) ~k' this coefficient is: 1 2.<kal't>-•l + i<ka>~<.,·•>+···+ z<ka>P'r-•l+cka.la,<i,-,l -a." • ► t,-• 2 terms In ¢1< ~l' the coefficient is: cka.fl'·•> +ckal'~-z1da.l +··· + <lalcp-•l - a""· r = k.. - l., k-1 at-a.- ► - a'1._., = 0 , if ne 1ther k nor 1 i s p . If k == p, the coefficient 1s: <1 ►<1>·•> -1) at> · ' = <lP·• -1) al>'·t, = 0. 1 5) Therefore, Thus an orthogonal set has been exhibited for the special case, n : l, t = 2. It will be noted that, in place (28) of the constant which usually begins such a set, there is here the function, r~-• {a;x), which is a constant equal to -a~~-~, by 2) of the above proof, except over the modulus M, where the value is zero. A complete generalization of this theorem does not exist. The statement is, however, capable of some exten- sion. The simplest case occurs when M 11 "' Mt-, ; for then Mt-, consists of only p elements, and the expansion of¢<~) in terms of f(Mt-, ;x) also has at most p terms. The following theorem may be obtained: Theorem 8.3: If tis even, the set: forms an orthogonal set where a is a fixed element of the For the proof of this theorem, lemmas which are extensions of Lemmas 1 and 2 are needed: · Lemma 3: The first p - 2 powers of f(Mt-, ;x) have sum zero over Mt_ 1 , while f ~-• (Mt _1 ;x) has s. non-zero sum. Proof: Mt-t consists of the multiples of a single element, b 1 , by the elements of TT. (29) Lemma 4: rP(M •x) = a,.., f(M ·x) where « 0 is 0 t-1' t-1' a fixed element of the GF(pt)$ Proof: By Th. 6.4: .,_, = ~/Jif"cMt_ 1;x). Clearly, ,id 0· = o, since fP has no constant term. The high- est power of x in: f .. <Mt-, ,~) -= (>t i,t-1 + Y.X pt•l+···) p is evidently pt- t • If a power higher than the first of f(Mt_,;x) were present in the expansion of fP, a higher power of x would occur. /3i. ; In the 0) Thus: i. -1- \ ; and /J I " .-v'" #1 • field-identity, fP( Mt-, ;x) =- 13 f (Mt-, ;x), choose a 1 value of x not in Mt- i , so that f (Mt-, ;x) # 0. Note: Then: It can easily be shown, though it is not germane to the present theorem, that: where (a 1 , ••• ,at_,) is the basis of Mt-i• (30) If and only if tis even, there exists an (at least one) element, c, of the GF(pt), such that cP-• = -1, when pis odd. Let ca 0 = 0t. .. With this remark, the main theorem is ready for proof: 1) The set forms a basis. This is proved precisely as in Th. 8 .2 • 2) Again, as in the previous theorem: and l=fi.cMt_,;x) I= o, r-• I i.. "'t-1 3) The reduction of L. 9>~ :I- 0, and L ~k ,!>1 = O, again proceed in the manner of Th. 8.2. This method is clearly not applicable to the case where p and tare odd since no arrangement of the type used can possibly result in a cancellation in the summation of L <fk ¢ 1 while retaining a fixed, non-zero, sum in the [ c-/>: . The lack of a (p - l)st root of -1 in these cases has an analogue in the necessary introduction of i = .J-=rin ordinary analysis. If the GF(pt) is imbedded in a • field where the (p - l)st root of -1 exists, then the expansion of periodic functions of the given modulus over (31) the original field is possible in terms of a set of ¢k similar to the above, but with coefficients from the super field. These remarks are not intended to indicate that no orthogonal set exists in the case where tis odd, but merely that the simple sets of Theorems 8.2 and 8.3, yielding Vandermonde determinants in terms of the f's, fail to exist. However, as previously remarked, the Vandermonde determinants offer the only method known to the writer to yield sets which for general values of p and tare guaranteed bases. Theorem 8.4: When tis odd, no orthogonal set, with a Vandermonde determinant in the f-functions, exists for the representation of functions with modulus Mt_,. Proof: Assume the existence of the set: Then the coefficient off ~ 1 after the reduction of r 2 ~- 2 which must be zero. lit - /Ji + OL .,_, "' 0, /31< -/Ji 0 (32) which is impossible when t odd for odd primes p. It might, however, be possible to obtain an orthogonal basis by taking a smaller period modulus. This problem may be illustrated by the case t = 3, n = 1, which will be worked out in detail. Lemma~: ~ If t 3, all powers of f(a;x) less than p2 - 1 have sum zero over M. Proof: Let the basis of M be (b,,bi)• = [ lY\ i = 0 if p - 1 f me+ nd. i, and thus the only cases of concern occur when j and i - j are both divisible by p - 1. implies that i its elf is di visible by p - 1. and consider ( This Let i = k( p - 1), k<p-1) ) , k > 1 I c~-1) ' In the numerator, [k(p - l)]t, p will appear k - 1 (33) times, asp, 2p, ... ,{k - l)p. appear { 1 - 1 ) + { k - 1 - In the denominator, p will times! or 1 ) k - 2 times. Therefore the binomial coefficient is divisible by p, and, thus, zero, in the field. The lemma then follows since the smallest power which is ?llot affected by the argument is given by k = p + 1, or i = p2 - 1. For this power, the sum over Mis clearly not zero by an argument of the same type as that given for the (p - l)st powers in the earlier theorems. Lemma 6: When t ""3, = - Proof: and the lennna follows. these lemmas give sufficient information to obtain the coefficients of interest in the expansion of ¢>k ci61 • When the product is reduced by the application of Lemma 6, the 1 coefficient of f P -, { a ;x) will be: (34) ttt_, ~ k 1 J. +~It-~ i- ·· ·+ k 1 ,i '1 -•♦ \'t__l)(/.3kp-1+···+.,..,1 A l>·•) +(_.:J_) al'- > 1 ;1'-1 which must be zero. Then or there exist elements, x, such that: It is easy to show that this equation is not, in general, solvable in the GF{p 3 ). In fact, considering the case a ~ 1, and multiplying by x: ~ and ,c. "3 t> l 0 or )l -)( t> 2. -x. " = 0 or xfl 2. )(. )> or )( x" -><..I>-)(. Subtracting, " : 0 : a. xP ,._ ){ , =-)( ► +it. : 0, which does not satisfy the original equation. That the condition [ ¢k # 0 is met by the system is clear if t,11k .,_ ,,dl in the above. The resulting coefficient is - a r:, i_ f> • The set has been chosen in such a way that if /3k ~ (3 1 , k ~ 1, the ¢k automatically form a basis. Furthermore, the equation xl>i - c~ _, - dx = 0 clearly has no (35) multiple roots in any superfield, and the roots in the splitting field of the equation form a modulus. Thus, if the GF(p3) is imbedded in the splitting field of: 2 J 1 3 x" -a.'p -P x., - a I> - P x -=- 0 , as a super field, an orthogonal set will be given by the indicated set of t6k if the /jk. are taken as the roots of the equation. The ¢k(x) in this case, considered as functions over the original field, are periodic functions wi t h the given modulus, and are then within the scope of Th. 3.5. is, therefore, a representation of the 'P1<<1t) There in terms of the f(a;x) with coefficients in the original GF(p3). Hence, in a sense, an orthogonal set has been demonstrated, although the actual form of the function has not been presented within the given field. Although the detailed work has been carried out only for the case t : 3, the generalization is immediate, and gives as the final major theorem: Theorem.§_&: In the case n ~ 1, p odd, there exists, in general, no orthogonal set of the type: t•I f pt•q, f ~ •' (a;){) .;bk = {;j /Jk with fjk in the GF(pt). However, if the original field is imbedded in the splitting field of: ( 36) t-1 X '1 = 0) and the ;dk chosen as the roots of this equation, the resulting set furnishes an orthogonal system with respect to the original field for the representation of periodic functions whose modulus contai_n s a, over the GF(pt). It may be noted that, in this case, no special results obtain for the even values of t. the defining equation becomes: X 1'>3 - x. _,z. If t -= 4, and a :a 1, l> - )( - ~ = 0 which, on raising to the pth power, yields: X Thus: or X X .,'Z. b =- - Xr , and .,4 -)( ►3 113 = Xt> 2 -X "2. ->C. I> =0. + >I. ► + ,c. h 3 Xr = - >(. Substituting in the original equation: or x = 0 is the only solution in the field. 9. Miscellaneous Results. In this section, several results will be presented which, while they have no direct connection with the (37) development of the main theory, are interesting in themselves or illustrative of that theory. a. The analogue of the sin nx system. In the set: a reordering may be made as follows: f .,•• cM, x) jo{~)= ~ <11:) =a.~-•+ ac.l'·.t f <M; x) +··· + f ►·'cM,x) y,,. c~>-= a"·' -1- 2.« ►• 2 fcM;x.) t··· + P... cM;x> The reordering is made so that fork I O., the new V~ is + , where 1 = in lT. It is 1 immediately apparent that the above set is identical with equal to the original the ¢,k Cx), ¢ and, also, that for k ~ 0., JP11. (>l) = jb, Ck K). Thus an analogy is established between the function., sin x, and ]b,<~), with the previously noted difference in the replacement of a constant term by the semi-constant, ft:>·•()(). The function, i/1. (X) , is, in general, neither odd nor even since it is used to represent all periodic functions of the given modulus. b. The number of periodic functions for given p and t. (38) The total number of distinct polynomials which may be written for a GF{pt), subject to the identity, xq ax, where q: pt , i~ qq. The question may be asked: How many of these are periodic? In the expansion of the singly periodic functions, there occur pt -, sunnnands, and the coefficients of each sunnnand may take on pt values; there are moduli. Thus., there are functions. Pt~-I -1 ( nt) ~t-, I" distinct periodic If the pt constant functions are discarded, the remainder may be grouped into classes of essentially distinct periodic functions. There are classes of associates, each with pt - 1 members, the classes of functions differing only in the constant term containing pt members. There then remain: (~t)pt-1 - pt (pt-,)pt =- The discussion has included all periodic functions. It is easily seen that, for a given modulus, the number of periodic functions whose modulus contains the given one is {pt)~ t-n , where n is the index of the modulus. A similar argument with respect. to division into classes may be made. A tabulation for the purpose of comparison is presented in the appendix. c. Derivatives of periodic functions: Again in analogy to the usual theory of periodic (39) functions where the derivatives are also periodic, the question may be raised: Are the formal derivatives of the periodic polynomials over a field of characteristic p also periodic? This question may easily be answered for Galois fields by consideration of the formal derivative of ri(Mn;x) which is i•r•- 1 (M 11 ;x)·Df(M,,;x). But Df(M 11 ;x) is simply the coefficient of x in f(M,,;x) since the latter is a p-polynomial . Call that coefficient~, and the result is i .(3 f i -, (M11 ;x ). expressed as Since all periodic functions may be i;. f ~ CM 11 ; x), it is clear that the formal L derivative of any per iodic function is also a periodic function with the same or a more inclusive modulus . It is pointless to discuss the formal integral of a periodic function since the integral of such a function is not merely indefinite to the point of a constant but to any polynomial consisting of terms of the type xnP . 10. Unsolved and Unf ini she d Pr obl ems. There follows a partial 11st of problems left unfinished or unsolved by this thesis. 1) Application of the results to higher congruences and diophantine equations. 2) Extension of the representation theory to more general fields of characteristic p. This will require (40) discussion of the convergence of infinite series of the functions used in this paper in finite sums. 3) Completion of the orthogonal set theory and possible application to algebraically complete fields of characteristic p. 4) Matric discussion of the operators involved and the application of matrices to the orthogonal set theory. 5) The possible existence of a more 'natural' set of functions than the present f-functions for description and representation of the properties of periodic functions. (The jtl1 function does not meet the condition since it does not exist for all fields and is inherently more complicated than the f's.) 6) The group-theoretic problem in which the functions are considered as endomorphism operators. closely allied to the ~irst and fourth. This problem is (41) APPENDIX In this appendix some specific and numerical examples will be given of the functions covered by the main body of the thesis, and a convenient tabulation of the simpler finite fields is presented. a. Table of finite fields The following table gives for all p, t : 2, 3, pt< 100, a complete tabular presentation of the GF(pt) in a double notation for convenience of both addition and multiplication . For each field, an irreducible polynomial of degree tis given in terms of x and as a vector , e .g., x2 ~ x + 1: (1,1,1). The left hand table gives the vector form of the residues of the basic irreducible polynomial which form the GF(pt), followed by the first primiti ve element occuring in the tabulation (designated by 'a') raised to the corresponding exponent. For simplicity both connnas and initial zeros are omitted from t he vectors, e.g . x + 1 is represented by 11 . The right hand table gives the exponent of 'a' followed by the vector representation. The two tables thus form a set of 'logs' and 'antilogs' for multiplication, while addition may be performed on the vector set by ordinary addition, reduced for each digit mod p. Starred elements are primitive. The vector representation also yields a convenient {42) method for choosing linearly independent sets. Any set of elements, each with a different number of digits in the tabulation, is linearly independent, and any linearly independent set is equivalent to a set of this type. These statements may readily be verified by considering the residues which the vectors represent. Square Fields x 2 + x+ l : (l,1,1) Exponent Exponent Vector 1 a3 1 10 {1-10 a 2 11 ~-11 a2 3 1 x2+ 1 •• (l,O,l) Exponen t Exponent Vector 1 a8 1 11 2 a4 2 20 10 a6 3 21 *11 a 4 2 *12 a7 5 22 20 a2 6 10 {t,21 a3 7 12 ~-22 a5 8 1 Vector 0 32: Vector 0 {43) x 2 + X + 1 : (1,1,1) Exponent E xponent Vector 1 a24 1 12 2 al8 2 33 3 a6 3 13 4 al2 4 40 10 al6 5 41 11 a20 6 3 ~:-12 a 7 31 13 a3 8 44 -i:-14 al7 9 34 20 alO 10 20 ~ 21 a21 11 23 22 al4 12 4 ➔123 all 13 43 -.1-24 al9 14 22 30 a22 15 42 -l!-31 a7 16 10 -l!-32 a23 17 14 33 a2 18 2 34 a9 19 24 40 a4 20 11 -ll-41 a5 21 21 - 42 al5 22 30 *43 al3 23 32 44 a8 24 1 Vector 0 \44:J x2 + 1 : ( 1., 0., 1) Exp. Vect. Exp. Exp. Vect. Exp. Vect. 1 a48 34 8 26 1 12 25 65 2 a32 35 a27 2 43 26 34 3 a40 *36 a41 3 42 -- 27 35 4 al6 ~ 40 a28 4 30 28 40 5 aB -·-- *41 al7 5 64 29 13 6 a24 42 a3 6 22 30 55 10 a12 43 a2 7 62 31 15 11 a22 44 a38 8 5 32 2 4j~l2 a 45 a45 9 53 33 24 ➔t-13 a29 *46 8 47 10 61 34 16 ~:-14 a35 50 a20 11 63 35 14 ➔t-15 a31 *51 8 37 12 10 36 60 16 8 34 52 a42 - ~- 13 26 37 51 20 a44 53 a9 14 33 38 44 -¾-'-21 al9 54 a39 15 23 39 54 22 a6 55 a30 16 4 40 3 23 8 15 *56 a43 17 41 41 36 24 8 33 60 a 18 25 42 52 25 8 18 a 19 21 43 56 Vect. 0 - ---- ,.., ...... ... 61 - · - ·· ,-. ~6 al3 ➔t-62 a7 20 50 44 20 30 a4 *63 all 21 32 45 45 '1~31 8 23 -~64 a5 22 11 46 66 *65 .. 8 25 23 31 47 46 a46 24 6 48 1 32 _ , .........,...8. -21 ,._ al4 33 ........ __,._ "-"--~·..---- ~- .... 66 -·- -~·-- {45) Cubic Fields . x3 + x + 1 : ( l, o, 1, 1 ) Vect. Exp. Vect. Exp. ~uoo ~no1 a2 a6 1 1*110 a4 a5 0 ! Exp. Vect. Exp. Vect. 4 110 10 5 111 2 100 6 101 3 11 7 l ~-·'"'' i I•:}111 x3+ 2x+ 1 : (l,0,2.,1) Exp. Vect. Exp. Exp. Vect. Exp. Vect. 1 a26 ~-112 all 1 10 14 20 2 al3 120 a4 2 100 15 200 *10 a 121 al8 3 12 16 21 -i~ll a9 ia22 a7 4 120 17 210 -r.-12 a3 ~00 al5 5 212 18 121 20 al4 ~01 a25 6 111 19 222 21 al6 202 a8 7 122 20 211 22 a22 ~-210 al7 8 202 21 101 :i,.oo a2 211 a20 9 11 22 22 ~-101 a21 ~~12 a5 10 110 23 220 102 al2 a23 ___ 11 112 24 221 110 alO 12 102 25 201 111 a6 a24 221 - -·-al9 -¾Z22 13 2 26 Vect. 0 -- ---·- --··~-.-- ------ ~20 , . .,., •·H• - - - . - I 1 t4oJ The theory applied to the fields, GF(3 2 ) b. and GF(33). In the notation of part a of the appendix, choose M~ =- TT . M -a. :. (11,101), Also choose F(O) "'O, F(l): l, F(2) = 2, and the doubly periodic function becomes: •• 0 l 2 10 11 12 20 21 22 F(x): 0 1 2 2 2 1 1 .1 2 • 100 101 102 110 111 112 120 121 122 2 0 l l 2 0 0 l 2 •• 200 201 202 210 211 212 220 221 222 2 0 0 l 2 2 0 1 X X ,o F(x ): X F(x): 1 If the first line only is considered, a periodic function over the GF(32 ) is represented with M1 :-. {11); M, =- 1T • The basic function for the GF(32 ) case is: x3 - (ll)2x = x3 - (20x) = x3 + (lO)x. The function actually represented is (12)(x3 + (lOx). The basic function for the GF(33) case is: (47) The function presented is: (222) [x9 _.. (22 )x3 + ( 102 )x]. Returning to the GF(32), f( (ll);x) = x 3 + (lO)x, r2( (11) ;x},. x6 -t- (20)x4 + 2x2. The orthogonal set, using the 1/t -functions, is: 1 ~ (i)"' f ((u);K) ::. X. 6 +(~O)X4- +-l.i.1. 11, Cr.)= f'(c11);><) + (11)3 f(c11>;x) + {11) 6 = x. 6 +(1o)x 4 ~(.Z.l)K 3 ~ i.xL+ (11)~ + (10) The f-functions, in terms of the -y., -functions, are: I= (10)(~+1/:~~> f ( <u); ~) = (u.)CY.,1 - '¥,') f ~(< u) pc) = ¥' 0, The values of the~ -functions for M, are: (48) 1/{ ~ ¥'i, ?,t,!£. ¥.z l/lf ~o~ ~¥Ii. it:~~ 0 0 (10) (10) 0 2 2 0 0 2 1 (20) 2 1 2 l 1 (10) (20) 2 2 (20) 1 2 2 1 -1 (20) (10) 2 l l l 0 0 0 X 0 Totals: Thus the orthogonal properties are verified for the ~-functions of the given modulus. c. The number of periodic functions for certain fields. Field Polynomials Periodic Functions Classes (Cf. 9b) 22 256 48 3 32 387,420,489 2916 40 52 8 .8 • 103 4 58,593,750 97,656 72 6 .6 • 1082 5,425,784,582,792 2,306,881,200 23 16,777,216 28672 511 33 4.4 • 10 38 99,132,767,304,831 141,214,768,240 The table displays the relative rarity of the periodic functions more clearly than the literal relation given in Section 9b. It also shows the large number of functions involved for all but the simplest fields and the rapid increase with the complexity of the field. l49J BIBLIOGRAPHY General References 1) Dickson, L. E. - Linear Groups - B. G. Teubner, 1901 2) Speiser, A. - Die Theorie der Gruppen von Endliche Ordnung - Springer, 1927; Dover, 1943 3) van der Waerden, B. L. - Moderne Algebra - Springer, 1931, 1937, 1940; Ungar, 1943 Polynomials over a Finite Field 4) Garlitz, L. - The arithmetic of polynomials in a Galois field - Proc. Nat. Acad. Sci. 17, ( 120-122) ( 1931) 5) ------ , Am. J. Math. 54, 39-50 (1932) - On polynomials in a Galois field - Bull. Amer. Math. Soc. 38, 736-744, (1932) 7) - On a theorem of higher reciprocity Bull. Amer. Math. Soc. 39, 155-160, (1933) 8) - On the representation of polynomials in a Galois field as the sum of an even number of squares - Trans. Amer. Math. Soc. 35, 397-410, (1933) 9) - On polynomials in a Galois fie.ld: Some formulae involving divisor functions. Proc. London Math. Soc. IIs 38, 116-124, {1934) (50) 10) Garlitz, L. - On certain functions connected with polynomials in a Galois field - Duke Math. J., 1, 137-168, (1935) 11) - On the representation of polynomials in a Galois field as the sum of an odd number of squares - Duke Math. J., 1, 298-315., (1935) 12) - A theorem on higher congruences - Bull. Amer. Math. Soc . ., 41, 844-846, (1935) 13) - On certain higher congruences - Bull. Amer. Math. Soc., 41, 907-914, (1935) 14) - On factorable polynomials in several indeterminates - Duke Math. J., 2, 660-670, ( 1936) 15) - Some formulas for factorable polynomials in several indeterminates. Bull. Amer. Math. Soc., 43, 299-304, (1937) 16) - Criteria for certain higher congruences Amer. J. Math., 59, 618-628, (1937) 17) - An analogue of the von Staudt-Clausen theorem - Duke Math. J., 3, 503-517, (1937) 18) - Some sums involving polynomials in a Galois field - Duke Math. 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