EXCHANGE 
 
 ^-Jj 
 
Z\)C Tantveratti? ot Ci^ica^o 
 
 ^'^CHA»:(33 
 
 nrr 
 
 lu 
 
 DNCERNING COMPACT KURSCHAK FIELDS 
 
 A DISSERTATI(3N 
 
 QUITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN 
 CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 DEPARTMENT OF MATHEMATICS 
 
 BY 
 
 VISHNU DATTATREYA GOKHALE 
 
 Private Edition, Distributed By 
 
 The University of Chicago Libraries 
 
 Chicago, Illinois 
 
 Reprinted from 
 
 The American Journal of Mathematics 
 
 Vol. XLIV, No. 4, October, 1922 
 
^be inmveretti? of (tbtcaoo 
 
 CONCERNING COMPACT KURSCHAK FIELDS 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN 
 
 CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 DEPARTMENT OF MATHEMATICS 
 
 VISHNU DATTATREYA GOKHALE 
 
 Private Edition, Distributed By 
 
 The University of Chicago Libraries 
 
 Chicago, Illinois 
 
 Reprinted from 
 
 American Journal of Mathematics 
 
 Vol. XLIV, No. 4, October, 1922. 
 
a^ 
 
 \^ 
 
 Or 
 
 u 
 
 €^f' 
 
 ^^ 
 
 .T»»* 
 
CONCERNING COMPACT KURSCHAK FIELDS. 
 
 By Vishnu Dattatreya Gokhale. 
 
 CONTENTS. 
 
 I. Introduction 298 
 
 1. Definition of a field, characteristic 298, 299 
 
 2. Algebraic closure, algebraic extensions 299 
 
 3. Definition of a KiirscMk field 300 
 
 4. Perfection, Kiirschdk's results 300 
 
 II. Compactness and Algebraic Extensions 302 
 
 1, 2. Compactness, compactness and perfection 302 
 
 3. Compactness and algebraic closure 303 
 
 4. 5. Hensel-Klirschdk fields, Ostrowski's results, applications 304, 306 
 
 6. Compactness and adjunction of algebraic elements 306 
 
 III. The Hensel-Kurschak Field 3cs 308 
 
 1, 2. The Hensel-Kiirschdk field X3 308, 310 
 
 3, 4. Perfection and compactness of Xg 310, 311 
 
 5. Subfield $R(g, x) of X^ 312 
 
 6. Adjunction of elements algebraic with respect to g 313 
 
 IV. The Properties p, A, P, cpt 314 
 
 1-5. Complete existential theory of these four properties 314 
 
 Abstract. — A Kiirschdk field is afield with a modulus ("bewertete Korper), in the sense 
 defined by Kiirschak in his memoir, "Ueber Limesbildung und die allgemeine Korper- 
 theorie" (Crelle, vol. 142, 1913). This modulus plays, in the general field, eesentially the 
 same role as the absolute value in the fields of classical analysis, viz., real number system, 
 complex number system, etc. Kiirschak proves that every Kiirschak field determines 
 (in the sense of isomorphism) a definite algebraically closed and perfect Kiirschdk super- 
 field, the smallest such superfield, being (in the sense of isomorphism) a subfield of every 
 such superfield. This definite superfield is the {smallest) algebraically closed and perfect 
 extension of the original Kiirschak field. 
 
 In the present paper the author sets up the notion compactness. This notion is 
 analogous to M. Frechet's compactness and to the J-compactness in E. H. Moore's General' 
 Analysis. It is a generalization of the following property in the point set theory: Every 
 infinite set of points in a bounded domain has at least one condensation point. He then 
 studies the properties of algebraically closed and compact fields, and compactness under 
 the adjunction of algebraic elements. Using Ostrowski's results he proves the theorem 
 that the smallest algebraically closed extension of a compact field is compact if, and only if, 
 it can be obtained by adjoining a single algebraic element. The last part of the paper 
 develops a complete existential theory of the four properties: (1) of characteristic other 
 than zero, (2) algebraic closure, (3) perfection, and (4) compactness. Out of the 2* = 16 
 possibilities 11 are shown to be existent and the remaining 5 non-existent. 
 
 297 
 
 529:^95 
 
298 GoKHALE : * Vdhderning Compact Kiirschdk Fields. 
 
 I. Introduction. 
 
 1.1. Field.— Following Moore,* H. Weber,t and Steinitz4 a field is 
 defined in the following manner: § 
 
 D "[ 1 Jt ^ C<jU . _1_ on'.5P'Pto:iS.1.2.6.6. \y on!5p>pto5P.3.4.5.7\ 
 
 where ^ = [[p] is a general class of elements p{pi, P2, etc.); +, X are 
 single-valued functions, + (pi, P2) is a definite element p denoted by 
 Pi + Pi, etc. ; and the properties numbered 1-7 are : 
 
 (Pi, P2, Pz) 
 
 (1) Pi + (P2 + Ps) = {pi + P2) + Ps ' 
 
 (2) Pl + P2 = P2+ Pi 
 
 (3) IP1P2)P3 = PliP2Pz) 
 
 (4) P1P2 = P2P1 
 
 (5) pi(p2 + Pz) = P1P2 + PiPz 
 
 Here (pi, p2, Ps) indicates that the equations (1-5) hold for every choice of 
 the elements pi p2 pz of the class '^. 
 
 (6) Pi-P2') '3l p^pi + p= P2 
 
 Here the meaning of the logical signs * ) ', 3,1, and ' will be clear 
 from the following reading of the property: — For every pi and p2, there 
 exists uniquely an element p such that pi + p •= P2. 
 Hence, 
 
 (60) 3 1 2^" : 9 : p : ) '. p -\- z = p-pz = z. 
 
 Here the notation 2^' reads "z belongs to the class *>|3." This unique element 
 z is called the zero eleTnent of the field. 
 
 (7) 3 p 7^ 2 : pi 7^ Z-P2 • ) • 3l p ? pip = P2, 
 
 to be read " There exsists an element p different from the element z (of 60), 
 and for every such element pi and every element p2 there exists uniquely an 
 element p such that pip = P2." 
 
 *E. H. Moore, "A Doubly Infinite System of Simple Groups" (Chicago Congress, 
 1893, pp. 208-242), p. 210. This paper will be referred to as M. 
 
 t H. Weber, "Die allgemeinen Grundlagen der Galoischen Gleichungstheorie " (Mathe- 
 matische Annalen, 43, 1893, pp. 521-549), p. 526; also "Algebra" (II edition, 1898), Vol. I, 
 p. 492. 
 
 t Steinitz, " Algebraische Theorie der Korper" (Crelle, 137, 1910, pp. 167-309), p. 172. 
 This paper wUl be referred to as S. 
 
 § For postulational definitions of a field see : 
 
 L. E. Dickson, "Definitions of a Group and a Field by Independent Postulates" 
 {Trans. A. M. S., Vol. 6, 1905, pp. 198-204), p. 202. 
 
 E. V. Huntington, "Note on the Definitions of Abstract Groups and Fields by Sets of 
 Independent Postulates" {Trans. A. M. S., Vol. 6, pp. 181-197), pp. 186, 191. This paper 
 also contains a bibliography. 
 
Gokhale: Concerning Compact Kiirschdk Fields. 299 
 
 This property can be easily seen to be equivalent to the following: 
 
 (7') 3 p^ z : 3u^' ^\j>') ' pu= p :p ^ z') -3 p' epp' = u]. 
 
 Also, we have 
 
 (7o) 3l u^' ^ p ' ) ' pu ='p': pi 9^ Z'pipx = z : ) : Pi = z. 
 
 This element u is called the unit element of the field, and the element (which 
 can be easily seen to be unique) p' associated with p in the second part of 
 7' is called the reciprocal of p. 
 
 Hereafter we denote the elements p of the class ''P of a field not by p 
 but by / (/i, /2, etc.). 
 
 1.11. Characteristic. — A field of characteristic p, in notation %^ is defi ned 
 in the following manner: * 
 
 D 1.2 %^ \ = \ % ' ^ ' 3 n ^ nu = z- p = the smallest such n. 
 
 By using the properties of the field it can be proved that in this case, 
 
 (1) p is a prime. 
 
 (2) / 5^ z : ) : n/ = z • ~ ' n = a multiple of p. 
 
 On the other hand, for every positive integer m and a prime p, there existsf 
 one and only one finite field, the so-called Galois field [_p^~\, with char- 
 acteristic p. 
 
 Fields without any such characteristic are said to be of characteristic 
 zero, in notation %^. Henceforth in the notation f^^ we shall understand 
 p to be an indefinite prime number. Thus every field is of the type '^^ or %'p. 
 
 1.2. Algebraic closure, algebraic extensions. — An algebraically closed 
 field 5, in notation f^"*, is thus defined: | 
 
 Z) 1 3 ^^'' ^ • '^ : 3 : 
 
 n-{hh, • • -,fn) : ) : 3 f ^ Up + /i/""^ +...+/, = ^. 
 
 An element j algebraic with respect to i^, in notation j^^ * is defined as: § 
 
 2)1.4 j^^'^':^ ■:{],%) :5:3 = -^n-(/i, - - - , fn) ' ^ ' 
 
 This unique integer n is called the order of j. Here x is an indeterminate, 
 and j belongs to a field %' containing % as a subfield, in notation, %' d %. 
 Steinitz has shown] | how to extend a field by the adjunction of an 
 
 * S, p. 181. Cf. also: J. Konig: — Einleitung in die allgemeine Theorie der algebraischen 
 Grossen (Leipzig, 1913), p. 408. Konig uses the terms "orthoid" and" pseudoorthoid" for 
 fields of characteristic zero and p respectively. 
 
 tM, p. 211. 
 
 X S, p. 260. 
 
 § S, p. 183. 
 
 II S. p. 197. 
 
300 GoKHALE : Concerning Compact Kiirschdk Fields. 
 
 algebraic element j. This extension of a field ^, in notation ^(j), is the 
 extension in the sense that it is the smallest and unique (in the sense of 
 isomorphism). He also shows how to get the extension which reduces 
 completely a given polynomial <p irreducible in ^. Steinitz also proves:* 
 
 Thm 1.1 e : ) : 3 W^' : ^ : g'^-r^"^ ' ) ' %" o %\ 
 
 This field %', unique in the sense of isomorphism, we denote by \^a: 
 the (smallest) extension of % having the property A. 
 
 Every field ^ has ojie and only one prime suhfield f '^o', Vo is isomorphic 
 with the rational number system or the integral number system taken 
 modulo p, according as ^5 is of characteristic or p. By absolute algebraic 
 field, in notation '^, we mean the algebraically closed extension of such 
 a prime field '^q J 
 
 A field i^', algebraic extension of ^, in notation % ^^^^ is thus defined as: § 
 
 D 1.5 W^'^' : ^ : (g', 5) : ^ : S' 3 g./'"' ' ) '/"'''- 
 
 Note: The negative sign denotes the absence of the property. Thus 
 the last implication should be read: Iff does not belong to %, it is algebraic 
 with respect to %. 
 
 1.3. Kiirschak Field. — A Kiirschak field, in notation 9^^, is thus defined: || 
 
 Z>1.6 9i^(l5;|| ||on5to.:-*'-°.1.2.3.4)^ 
 
 where i5 denotes a field, 51"^®*^-° the class of all real non-negative numbers, 
 and the properties 1-4 of the single-valued function 1 1 1 1 (which will here- 
 after be called the modulus), are: — 
 
 (1) /=z-~- 11/11 = 
 
 (2) IIMll = |l/illll/.|| .-. \\«\\ = l\,f f. 
 
 (4) 3/:,:||/||^0-|l/||^l • 
 
 Hereafter we designate the elements of 9t not by / but by k{ki, k^, etc.). 
 
 1.4. Limit, Perfect Extension. — A limit k oi a sequence ki, k^, ••• in 
 notation {kn}, is thus defined: Tj 
 
 D 1.7 Lkn=k\=\{k, [kn]) ^L\\k-kn\\ = 0. 
 
 * S, p. 287. 
 t S, p. 180. 
 
 I S, p. 199. 
 § S, p. 198. 
 
 II J. Ktirschdk, "Uber Limesbildung und allgemeine Korpertheorie" {Crelle, 142, 
 1913, pp. 211-233), p. 211. This paper will be referred to as K. 
 
 UK, p. 222. 
 
GoKHALE : Concerning Compact Kiirschak Fields. 301 
 
 As an immediate consequence of the definition, 
 
 Thm 1.2 Lkn= k-LK= k' :) : k = k' - \\k\\ = \\k'\\ = L\\kn\\. 
 
 n n n 
 
 The usual properties of limits, for instance: if a sequence has a limit, 
 every subsequence has the same limit and conversely; the sum of limits 
 equals the limit of the sum; etc., follow in the usual manner. 
 
 A Cauchy sequence {kn}, or {kn} satisfying the Cauchy condition: in 
 notation {kn}"-" : is: 
 
 D 1.8 {kn}'-'\= i {kn] :^\e :) :3ne-^-n>ne.) .\\kn- knA\ <e. 
 
 Here, as further, e is a real positive number. As an immediate consequence, 
 
 we have 
 
 Thm 1.3 3Lkn ' ) " {A;^)^-''. 
 
 n 
 
 The converse however is not true; e.g., in the field of rational numbers 
 with the absolute value as modulus, not all Cauchy sequences have rational 
 limits. A field dlo or a subclass 9?o (not necessarily a field) of a field 9? for 
 which the converse holds is called perfect: in notation dio. Thus the defini- 
 tion of perfection is: * 
 
 2)1.9 9??:=i9?o''^:3 : {konV'" ' ) ' 3 ko ^ L kon = ko; 
 
 n 
 
 in this definition (as also further, where necessary) we denote the elements 
 of 9^0 by ko{koi, fco2, etc.). 
 
 By a method analogous to G. Cantor's in building up the real number 
 system from the rational numbers, Kiirschak has shown f how to extend a 
 Kiirschak field so as to make it perfect. The smallest such extension 
 (unique in the sense of isomorphism) of aJR we designate notationally by 
 ^^ : the (smallest) extension of di having the property P. 
 
 Given a 9?^ and j^^^, Kiirschak defines || j || so that 9?(j) is a Kiirschak 
 field: X in notation 9J(j)^. Defining in dl^ the modulus of every element 
 algebraic with respect to di in the same manner, we have 9?^. Making 
 this perfect, we get 9?/p, in the sense (9?a)p. Kiirschak then shows§ that 
 di^p is algebraically closed. If we start with any Kiirschak field 9f, we 
 
 * K, p. 228. 
 
 t K., p. 228. 
 
 JK, p. 245. Kiirschdk defines \\j\\ to be ||/„|1^'», where /„ is the /„ in D 1.4. A. 
 Ostrowski in his "Tiber sogennante perfecte Korper" {Crelle, 147, 1917, pp. 191-204), 
 p. 196, and "Uber einige Losungen der Functionalgleichung <p{xy) = <f>(x)<p(y)" (Acta 
 Mathematica, 41, 1918, p. 271-284), p. 280, has shown that this is the only possible 
 definition of || j|l. 
 
 These papers will be referred to as O2, O3 respectively. 
 
 § K, p. 251. 
 
302 Gokhale: Concerning Compact Kurschak Fields. 
 
 must first make it perfect and then follow this process. Thus: 
 
 Kiirschak's final theorem is thus: f 
 
 Thm 1.4 ^ : ) : ^Rifp '^ • ^'^-^-^^ ' ) ' ^' ^ 9?p^p 
 
 This perfect and algebraically closed extension of 9^ we shall denote by 
 ^AP or 9^p^. 
 
 II. Compactness and Algebkaic Extensions. 
 
 2.1. Compactness. — In classical point set theory we have the theorem 
 that every infinite set of points in a bounded domain has at least one con- 
 densation point. A similar property for a general class for which the limit 
 function exists is defined by Frechet.| Classes having this property are 
 said to be, according to his nomenclature, compact. The following defini- 
 tion of compactness in the case of a Kurschak field is naturally suggested 
 as analogous to this definition of Frechet and the definition of J-compactness 
 in Moore's theory. § 
 
 A subset (not necessarily a field) of a Kurschak field is said to he compact 
 if every infinite sequence of elements of the set such that the (smallest) upper 
 bound of the modulus is finite, contains at least one Cauchy subsequence with 
 its limit element in the set. 
 In notation: 
 
 ^ ^'^ • ) • 3 {h, {/l^}P^--°°-*'^-) ' L hn^ = ^0. 
 
 Here as in 7) 1.9 9^o is a subset not necessarily a field of a Kiirschak field 
 di; and {rim} is a properly monotonic increasing sequ£nce of positive integers 
 n. This property denoted notationally in the above definition as pr. mon, inc. 
 is thus defined: 
 
 D 2.11 {ri,„}P^-'"»°'°''- : = i {nm} : ' : mi > mz • ) ' n^, > nm,- 
 
 2.2. Compactness and Perfection. — We now prove that: 
 Thm 2.1 . W^'-)'diE- 
 
 * In his theses, "Uber einige Fragen der allgemeinen Korpertheorie" {Crelle, 143, 
 1913, pp. 255-284). p. 284, A. Ostrowski finds under what conditions this step is necessary. 
 He also shows (p. 260) that the algebraically closed extension of Hensel's p-adic numbers 
 is not perfect. This paper will be referred to as Oi. 
 
 tK, p. 251. 
 
 t M. Frechet, "Sur quelques points du calcul fonctionnel" {Rendiconti del Circ. Math, 
 d. Palermo, 22, 1906), p. 6. 
 
 §E. H. Moore, "Lectures on Matrices in General Analysis" (University of Chicago, 
 1919-1920). 
 
Gokhale: Concerning Compact Kilrschdk Fields. 303 
 
 Proof: Consider a Cauchy sequence {kon}. Since it is a Cauchy se- 
 quence, 
 
 e :) :3nc^n> ne') ' \\ hn — hn, || < e. 
 
 Therefore for such ann, \\ kon \ \ Hes between | 1 1 kon^ 1 1 — ^ I and \ \ kon, 1 1 + «• 
 
 Hence B 1 1 fcon ! | < °o • Therefore by compactness we have a subsequence 
 
 « 
 
 having a Hmit in 9?o- Hence the original sequence has the same limit in 
 dlo. Hence the theorem. 
 
 Note: The converse of this theorem, however, is not true. See Thm 
 2.3 below. Compare also theorems 3.3 and 3.4 in the next section. 
 
 2.3. Compactness and Algebraic Closure. — 
 
 Thm 2.2 di^'-'^'-a^O :) : 3k^ \\ k \\ = a. 
 
 Proof: The theorem is obvious when a = or 1, the corresponding k 
 being z and u respectively. When a is neither the proof is as follows : 
 
 By D 1.6 property 4, there exists an element whose modulus is neither 
 nor 1 , Let this element be denoted by ko and let its modulus be ^. 
 
 By the theory of the real number system 3 {r„} ^ jL ^'■" = a, where 
 
 n 
 
 {r„} is a sequence of ordinary rationals r. Hence by 9^"*, 
 3 {k'o''}-B\\k^''\\ <oo. 
 
 Therefore by di'^\ 
 
 3 (k, {nm}'''-'^^''-^'-) 3 L t-n. = k 
 
 m 
 
 and I! k \\ = L\\ kl"m \\ = L ^'""m = a. 
 
 m m 
 
 Using Moore's results,* Steinitz proves: f 
 
 Thm 2.3 Z^" 5^ 2 : ) : 3 m ^/^^-i - u = z. 
 
 We shall use this theorem to prove: 
 
 Thm 2.4 9^A-«p* ' ) ' 9^°. 
 
 Proof: We shall give a direct proof of a contrapositive of the theorem, 
 viz., 
 
 ^p-A . ) . S)J-cpt^ 
 
 Every 91^'^ contains '^p, the absolute algebraic field characteristic p. 
 Consider { k^ } ''''«*^'=*- ^''. By theorem 2.3 we have : 
 
 n:):\\kn\\ = or || ^„ [h = i, 
 
 where q = p"^ — 1 in theorem 2.3. Hence 5 || ^n 11 = 1 .'. < oo. But 
 
 * M, p. 220. 
 tS, p. 251. 
 
304 GoKHALE : Concerning Compact Kilrschak Fields. 
 
 this sequence cannot have a Cauchy subsequence; for, otherwise, 
 
 e : ) : 3 (ni, n^)^'"^''^ II ^n^ - ^nJI < 2e, 
 
 viz., ni, n2 each greater than ne in D 1.8. But from theorem 2.3 and the 
 fact that ni9^ n2')' \\ K, \\ 9^ \\ kn^ ||, we have 
 
 ni9^ n2')'\\ kn, - kn,\\ = 1. 
 
 Taking e < 1/2 we get a contradiction. Thus this particular sequence has 
 no Cauchy subsequence. Hence the theorem. 
 We have further: 
 
 Thm 2.5 dt"""") -3 n' II 7m II 5^ 1. 
 
 Proof: Here also we prove the contrapositive, viz.. 
 
 di^n-)-\\nu\\ = 1:) : ^-^p*. 
 
 Since II nw II = 1 (n), we have the characteristic zero, and || rw || = 1 (r) 
 where r is any ordinary rational different from zero. Consider { r„ } ^*^"'''*. 
 
 We have 5 II r„2t II = 1 .'. < 00. 
 
 n , 
 
 Now the sequence {rnu] cannot have any Cauchy subsequence; for 
 otherwise, 
 
 e:) :3 (ni, na)*^"""* ^ || rn,u - rn,u \\ < 2e, 
 
 but since niP^ n^') ' r^u — rn^ = {rn, — rn,)u 7^ z')' \\ rn,u — Vn^u || = 1, 
 taking e < 1/2 we get a contradiction. Hence the theorem. 
 We have also: 
 
 Thm 2.6 dt'"' • ) • (the first n^\\nu\\9^ 1)^'^^ 
 
 Proof: For otherwise if w = nin2, where ?ii ^ ng < n, nu = nin^u 
 = {niu){n<2.u). Hence || tiw |1 = 1 which contradicts the hypotheses. 
 
 2.4. Hensel-Kurschak Fields, Ostrowski's Results. — A Kiirschak field 
 where the property 3 of the modulus in D 1.6 is replaced by the stronger 
 property: 
 (30 II /i + /2 II ^ greater of (|| /i ||, || f, \\) {f„ f,) 
 
 is called a Hensel-Kurschak field: in notation ©. Thus: 
 
 D 2.2 (S=(g;|| ||on!5to5l'eal-ao.l.2.3'.4), 
 
 where the properties 1, 2, 4 are those in Z) 1.6, and 3' is defined above. 
 Hereafter we denote the elements of <S by h{li\, hi, etc.); the class of all 
 Hensel-Kurschak fields will be denoted by H, and the superscript H will 
 denote the property of belonging to this class. These notations will be 
 used only when it is necessary to emphasize the fact that the Kiirschdk 
 field under consideration is a Hensel-Kurschak field. 
 
Gokbale: Concerning Compact Kurschak Fields. 305 
 
 The modulus of a Hensel-Kiirschak field, which, when necessary, we shall 
 call Hensel-Kiirschak modulus, is what Ostrowski* calls a non-Archimedian 
 modulus: in notation: || ||non-arch_ jjg g^jg^ proves: 
 
 Thm\ 2.7 II ll^o'^-^'^i': '^ : n' ) ' \\nu\\ ^ I. 
 
 Thmt2.8 \\h\\> \\h,\\ :) :|Ui + A2|| = || Ai ||. 
 
 A modulus which does not obey this stronger property 3' is called 
 Archimedian: in notation: jj ||^''^ 
 
 A. Ostrowski has investigated real-valued solutions of the functional 
 equations : 
 
 (p{xy) = (p{x)(piy) 
 
 (^ ^yatlonal 
 
 <p{x-\- y) ^ ip{x) -f (p{y)\ 
 
 His conclusions, modified for the Kiirschak modulus by the fact that the 
 modulus is non-negative, are: 
 
 rAm§2.9 ^0:): (1) r 5^ 2 ' ) '• || r || = 1; || k || = 
 
 or(2)r-)-||r|| = jrl" 0<p^l 
 
 or (3) r • ) • II r II = c"^-^^ < c < 1, pP'^''"^ 
 
 Here 0(2?, r) is the order of r with respect to p, defined, following Hensel || as: 
 
 Z) 2.3 I ^ = -rP" ' (^' ^yvimetop I . ) . o(^^ r) 
 
 V 
 
 In the above theorem, the modulus is Archimedian in case 2, non-Archi- 
 median in the other two. For n = 2 in theorem 2.6, there does exist a 
 field with Archimedian modulus, viz., glcompiex^ ^j^j^ ^^^ modulus as defined 
 in the second part of theorem 2.9. If we denote this field by 5Ip, and define 
 equivalence: in notation ~ : as isomorphism between Kurschak fields 
 which is preserved under the limit process, we have: 
 
 Thm 2.10 
 
 P1-P2 
 
 ■■^■■-^'--" iitm- 
 
 Ostrowski prov 
 
 es:1I 
 
 
 Thm 2.11 
 
 
 ^-"■)'3p^di c5r. 
 
 and finally,** 
 
 
 
 Thm 2.12 
 
 
 9fl-~a.P.A..y^ ^5JcompIex_ 
 
 * O3. p. 273. 
 
 
 t03, p. 273. 
 
 
 
 JOs.p. 274. 
 
 
 
 § O3, p. 276. 
 
 
 
 11 K. Hensel, ' 
 
 'Theorie der algebraischen Zahlen" (Leipzig, 1908, Vol. I). 
 
 IfOa, p. 281. 
 
 
 
 ** O3, p. 282. 
 
 
 
306 GoKHALE : Concerning Compact Kiirschak Fields. 
 
 2.5. Deductions from Ostrowski's Results. — rrom theorem 2.7 the 
 Archimedian or non-Archimedian character of the modulus depends upon 
 the moduli of the elements of the prime subfield. Hence we have: 
 
 Thm 2.13 9^' 3 9? : ) : 9^^^"^^ ' - * 9e'^^-^>. 
 
 Here the two parentheses go together. 
 From theorem 2.12 we have 
 
 nm2.14 ^^-^•^■^••)-9l«p*. " 
 
 We shall now prove an important theorem to be used in the sequel: 
 viz., compactness is not extensionally attainable* To be more precise, 
 
 Thm 2.15 9fi-oi>t.p • ) • - 3 9fi'^^.cvt^ 
 
 Proof: Since the field is perfect every Cauchy sequence has a limit 
 element in the field. Want of compactness, therefore, is due to the fact 
 that there exists an infinite sequence which has no Cauchy subsequence. 
 Now every extension preserves the moduli; this fact will therefore persist 
 in every extension. Hence no extension will make the field compact. 
 
 We show in section 4 theorem 4.10o that ^~pf)', where ® is the Hensel 
 field of p-adic numbers. Since in this case compactness is not extensionally 
 attainable (in virtue of theorem 2.15), we have, using theorems 2.9 and 2.4, 
 
 Thm 2.16 di^- "^P* : ~ : 91 ~ glcompiex^ 
 
 2.6. Compactness and Adjimction of Algebraic Elements. — An algebraic 
 element of the first kind: in notation algi : is thus defined : f 
 
 p 24 •^''''' :^ ^ I U,m-' :f'' : <p'-'^-™^"'-^'- ■<pU) = z 
 
 Note: Here the dot in the last brackets stands for the argument of the 
 function \_x — j | a:]: read "a; — j as a: varies." 
 
 An algebraic element, which is not of the first kind is said to be of the 
 second kind: in notation: alg2. Similar definitions hold for algebraic 
 extensions of the first and second kind. 
 
 Ostrowski proves: J 
 nm2.17 9?^-i^^^^-)-9?0r. 
 
 We prove the analogous theorem: 
 Thm 2.18 g^cpt.jaig, n.y ^(^jyvt^ 
 
 * A propertT/ is said to be extensionally attainable when there exists an extension which 
 has that property. Compare E. H. Moore, "Introduction to a Form of General Analysis," 
 Yale University Press, 1910, pp. 53, 54. 
 
 tS, p. 231. 
 
 t Ox, p. 275. 
 
Gokhale: Concerning Compact Kiirschdk Fields. - 307 
 
 Proof: Let m be the order of j and fx an upper bound of the modulus of 
 an infinite sequence {kn, ij"*"^ -{-■•• + kn, m\ n} oi elements of 9^(j). We 
 have to show that this sequence has at least one Cauchy subsequence with 
 a limit element in di(j). 
 
 Let us denote this sequence by {io, n] and let the m conjugates of io.n 
 be denoted by ii, „, 22, n, • • •, im, n', *i, n = io, n (n). Let the m conjugates 
 of j be denoted by ji, j-z, " • , jm', j ^ ji- 
 
 In the normal extension* of dt which contains 9?0) we have for every 
 n the m linear equations: 
 
 kn, ijr' + • • • + ^-n. .. = ^1. n 
 kn, ij^r' + • • • + A;.,, m = *2, « 
 
 /<^n. ij",!. ^ + ■ • ■ -^ kn. m = im, n. 
 
 Since j is of the first kind, the determinant of coefficients involving j and 
 its conjugates is not the zero element. Hence we can solve for the A:'s 
 and get kn, r {r, n) as linear functions of the i's with coeflScients independent 
 of n. Also !| ?r, n 11 = II *o. n \\ (w, r). Hence 5 || A;„, r !| < 00 (r). Consider 
 
 n 
 
 now the sequence [kn, 1}; by Dt*'^* and the result proved just now, this 
 sequence has at least one Cauchy subsequence with a limit element in 9?. 
 Take one such Cauchy subsequence and the corresponding subsequence of 
 [id, n\, say [i'n]. Denote the coefficients of the different powers of j in 
 this sequence by prime letters. As above we can prove that [k'n, 2} has a 
 Cauchy subsequence. Take the corresponding subsequence of [i'n] which 
 is itself a subsequence of the original sequence. Continuing this process, 
 after a finite number of steps, viz., m steps, we get a subsequence of the 
 original sequence such that the coefficient sequences are all Cauchy se- 
 quences. Let this final sequence be denoted by [inh Then if 
 
 in = kn, li"*-^ + \-kn,m (w), 
 
 we have: 
 
 nr7l2 : ) : II in, - in, \\ ^ F(|| K, 1 - K^ 1 II, - ' II K, m - kn„ m |i) 
 
 ■ F(llilh-S •••, liill, 1), 
 
 where gr denotes "the greater of." Hence this subsequence is a Cauchy 
 sequence. Using theorems 2.1 and 2.17 we see that the limit element is in 
 $K(j). Hence the theorem. 
 
 . An algebraic extension is said to be finite with respect to the original 
 field: in notation fin : when it can be obtained by adjoining a finite number 
 * S., p. 207. 
 
308 • Gokhale: Concerning Compact Kurschdk Fields. 
 
 of algebraic elements to the original field : * 
 
 D 2.5 ^'«-^ ! ^ I m', $K) : ^ : 3n-{h,k, ■ ' ^iO^'^'^ ^ ^' = ^^(ii, • • ^jn). 
 
 Steinitz proves: f 
 
 Thm 2.19 di- (ii, • • • , jnr^^ ^' : ) : aj^'^' ^ ^ 9^0') = 9?(ii. ' • ' , in). 
 
 Ostrowski proves : t 
 
 TAm 2.20 di'' :) :dif')- diT' ''■ 
 
 Using theorem 2.16, he then proves§ a theorem, which, in view of 
 theorem 2.19, we can state as: 
 
 Thm 2.21 $R^ : ) : 9?r ~ • ^j'"" ' ' 9^0*) = ^a- 
 
 We prove the analogous theorem: 
 Thm 2.22 di'"" : ) : dlT ' ~ * 3/'^^ ^ di(j) = ^^. 
 
 Proof: Since compactness implies perfection, from 2.21 we have 
 
 The other part of the theorem follows from theorems 2.20 and 2.18. 
 
 III. The Hensel-Klrschak Field 36^. 
 
 3.1. The Field BEj. — Given a field g we build a system: || 
 
 D 3.1 I, ^ [all <p-^^'-^' : 3 : 3^5 z' < i^ ' ) ' <p(i) = z]. 
 
 Here 3 = [f} is the class of all integers i. 
 
 Addition of elements of the system is the usual addition of functions: 
 
 D 3.2 ^i + ^2 = (^i(i) 4- ^2(^) I i) {(pi, (pi). 
 
 Multiplication is thus defined: ^ 
 
 D 3.3 ipnp2 = ( X) <P\{i\)(pi{i2) I *) (^1, <P2). 
 
 <l,'<2lii + i2=i 
 
 * S, p. 199. 
 
 t S, p. 220. 
 
 JOi, p. 281. 
 
 § Oi, p. 284. 
 
 11 In this definition it is obvious that when <p is not the zero function the i^'s, in the 
 definition have a maximum. We denote this maximum by j'(^) or v^ov more simply by v 
 when it is obvious to which element it belongs. Also in this case (p{v) 7^ z. In case ip is 
 the zero function such a maximum does not exist; we denote this symbolically by regarding 
 V as =0 . 
 
 \ It is to be noticed that the summation in this definition is finitely non-zero, and 
 hence no questions like convergence, etc., are involved; for by D 3.1 both i\ and 12 have finite 
 lower bounds for which the functional values of <p\, V2 respectively are non-zero, and since 
 i\ •\- ii = i the upper bounds too will be finite and for values greater than these the products 
 of the functional values will be zero. 
 
GoKHALE : Concerning Compact Kiirschdk Fields. 309 
 
 We shall hereafter designate the system in D 3.1 with addition and 
 multiplication as defined in 3.2 and 3.3 by Xj. We also introduce a notation 
 for a special set of elements of 36s. 
 
 D 3.4 di = (u at i and z elsewhere) (i). 
 
 We now prove the theorem that this system is a field : 
 
 Thm 3.1 g • ) • 3ef. 
 
 Proof: The properties 1-6 and hence 6o in D 1.1 are easily seen to be 
 satisfied from the fact that ^^ is a field ; the zero element is in this case the 
 zero function (p(i) = z (i). As for the property 7 we shall prove its equiva- 
 lent 7'. The first part of this property is obvious; also the element do 
 satisfies the first condition for the unit element, and in fact also the second 
 condition, that is, 
 
 (p ^ 2*5 • ) • ^(^' ^ ipip' = 5o. 
 
 Consider the v belonging to this (p (footnote 3.1) ; then an effective function 
 (p' is the function (p' with (p'{i) = z f or ? < — v, (p'{— v) = ul(p{v), and 
 <p\i) f or ?■ = — V -{- s (s > 0) as the elements of i^ obtained by the sequen- 
 tial solutions f or 5 = 1, 2, • • •, of the equations: 
 
 X; <p{v +'ii)<p'i- V + ^2) = 2 {s> 0). 
 
 Note 1: If we denote symbolically hi by a:', the elements of the field 
 BEj can be symbolically expressed as infinite series J^ifiX\ where /i = (p(i) (i). 
 Addition and multiplication can then be regarded as the corresponding 
 formal operations on the infinite series. We shall, therefore, hereafter take 
 these infinite series as the elements of 3tj and operate with them since this 
 procedure is formally more convenient. We shall sometimes denote the 
 range of summation of the index i as from i' to oo, i; being the integer 
 defined in footnote 3.1. 
 
 Note 2: We now identify (in the sense of isomorphism) this field with 
 the field ^(a;) of Steinitz, where x is transcendental* with respect to in- 
 putting the various powers of x in one field into correspondence with the 
 same powers of x in the other, and the elements of ^ with the same elements 
 of % in the other, we see that ^5 d ^^ [a;], where ^^[a:] is the integral domain 
 obtained by adjoining the transcendental x. Since the field ^(a^) is obtained 
 from this integral domain by the process of building up of quotients, f we 
 see that 365 3 %(.x). On the other hand it is obvious that 3Cj c ^(x). 
 Hence the result required. • 
 
 * S, p. 184. 
 t S, p. 178. 
 
310 Gokhale: Concerning Compact Kilrschak Fields. 
 
 We shall therefore now speak of this process of forming 365 from f^ as 
 the adjunction of a transcendental element x to the field %. 
 
 3.2. The Hensel-Kiirschak Field ^Cj..— Let ^ be a real number < ^ < 1. 
 We now define the modulus for the field H-^: 
 
 2)3.5 Ih-'HNO; <p9^zh')'\\^\\^^^'.. 
 
 The modular properties 1 and 4 of D 1.6 are obvious. The property 2 is 
 also obvious in case one of the factors is the zero function. In the other 
 cases, 
 
 V{ipnp2) = V((pi) + V{<p2) {(Pi, (P2). 
 
 Hence 
 
 !| <P1<P2 II = r*"*^^ = ^K^)+Kv.) = ^K^)^.(..) = II ^, II II ^2 ||. 
 
 Thus 2 is proved completely. 
 
 Again, for every (pi and (p2, v{<pi-\-(p2)= the first i for which (pi(i)-\-(P2{i) 
 is not zero (if there is any such, otherwise ^1 + ^2 = 2 and the property 
 3' is obvious) ; thus 
 
 v{(pi + (P2) = the smaller of v((pi), v{(p2). 
 
 In this case since < ^ < 1, we have: 
 
 ^1-^2 : ) : II ^1 + <P2 II = k"''^'-^-' ^ gr{^''V\ e"^') .-. ^ f(II <Pi II, II <P2 ||). 
 Thus 365 is a Hensel-Kiirsqhak field (cf. D 2.4), denoting by 36^ the field 3Ej 
 with the modulus as defined in D 3.5. 
 
 Thm 3.2 S • ) • 3£f. 
 
 Hereafter we shall generally denote the elements of 36,^ by h{hi, ^2, etc.). 
 The subscript ^^ in 36 ^ shall be dropped unless it is necessary to bring in 
 evidence the field from which 365 is formed. The elements of <5 in 36g 
 (by isomorphism) will, when it is desirable to bring that fact in prominence, 
 be denoted by /(/i,/2, etc.). 
 
 3.3. Perfection of 365. — We now prove the theorem: 
 
 Thm 3.3 S • ) • aef . 
 
 Proof: Consider a Cauchy sequence {A„}. From property 3' of the 
 modulus either X„ 1 1 ^n 1 1 = in which case the limit of the sequence is z or 
 
 3{no,v)^\\hn\\ = r = -^ll^nll in> no) 
 
 for otherwise if e < X 1 1 ^^ 1 1 which exists, and a = jL 1 1 A„ 1 1 — e, 
 
 n > Tie : ) : 3(wi, ng) ^ || hn, \\ 7^ \\ K^ \\')'\\ h^ - h„, \\ 
 
 = 9ri\\hn,\\,\\hn,\\)>a 
 
Gokhale: Concerning Compact Kiirschdk Fields. 311 
 
 where n^ is the Ue for the original Cauchy sequence, and this is impossible. 
 With this V and A„ = IIi""/n, i x' (n), we have 
 
 3no:^ :n> no') ' fn, i = /no. i (1) 
 
 for otherwise n \) :3 (ni, n^) > n^\\ hn^ — hn^ \ \ ^ ^""^ which is not true 
 as {hn} is a Cauchy sequence. Similarly, 
 
 i :) : 3no, i :^ : n > no, i' ) ' fn, ^i = fn,, ^ v+i- (2) 
 
 If we now define h = J^i'"fiX\ where fi ^ /„„_ ._^, i {i ^ v) in (1) and (2), 
 we see that i :) : \\ h — h„^^ < !l = ^"'^^ and so h = Lhn. 
 
 n 
 
 3.4. Conditions for Compactness of 36,ij. — Though every X as we have 
 seen is perfect, it is not necessarily compact. The necessary and sufficient 
 conditions for compactness are given by the following theorem: 
 
 Thm 3.4 If • ~ • %'">' • ~ • 5«"'t^ 
 
 Proof: Since/ ?^2')*||/|| = 1, Lfn = h' ) ' h^; therefore if we have 
 
 a Cauchy sequence of elements of % its limit element, when it exists, must 
 be an element of %. Take now an infinite sequence of elements of ^. 
 1 is an upper bound of the modulus; therefore by 3^1"* this sequence has a 
 Cauchy subsequence with a limit element. This limit element by the 
 above is in 5. Hence, 
 
 It is also obvious that g*^""^ • ) • ^<=pt^ f^j, jj^ ^j^jg (,^gg every infinite 
 sequence of elements of ^ must have at least one of its elements repeated 
 an infinite number of times. 
 
 Next suppose g'"*; consider {hn} ^ B \\ hn\\ < ^ , say < ^'. Let 
 
 n 
 
 hn^ Y,fn, i^:' (n). Then jn, i= z {n, i < t). Since every sequence in % 
 has 1 as an upper bound of the modulus, and hence by S*'^* has a Cauchy 
 subsequence with a limit element the sequence {/„, <} has a Cauchy sub- 
 sequence with a limit element. Take the corresponding subsequence of 
 {hn}. If the coefficients in this sequence be denoted by letters with the 
 upper subscript 1, we have a sequence whose terms begin with a power 
 not less than t and the coefficients of x* form a Cauchy sequence. Also 
 since every non-zero element of ^ has the modulus 1, after a finite number 
 of terms, the terms in this coefficient sequence are identical. The coefficient 
 sequence {fn^t+i} has similarly a Cauchy subsequence. Take the corre- 
 sponding subsequence oi {hn}. If we continue in this manner, and denote 
 the successive distinct subsequences of {hn} obtained in this manner by 
 lei, lei, '--Ah^'}, --^wehave 
 
 {hn} z> {h'i^ D {e-^^M (r>0) (1) 
 
312 Gokhale: Concerning Compact Kilrschdk Fields. 
 
 and 
 
 r> 0:) :3nr:^:m> r-n> nr-i <t-\-r\):ffi = jn^, i- (2) 
 
 If this series of subsequences has a last term, say {hn^], then its coefficient 
 sequences are all Cauchy sequences and it can be easily proved that l^*"^} 
 is itself a Cauchy sequence, and therefore by Xf, with a limit element in Xj. 
 
 If the series of subsequences has no last term, take the subsequence of 
 {hn} formed in the following manner: from each term of the infinite series 
 select the first element which does not belong to the next term in the 
 sequence (such elements do exist, since the terms of the infinite series are 
 distinct). By using (2), this sequence can be seen to be a Cauchy sequence 
 and therefore by the perfection of 36 j, with a limit element in 36 j. 
 
 Finally, to prove ^""^^ ' ) ' g^^te. y^g prove this by proving directly 
 the contrapositive, viz., 
 
 Cfclnflnite • \ . ^— cpt ^ 
 
 For consider {fnV''''^''; then 5 ||/n || = 1. If this sequence has a 
 
 ■Cauchy subsequence, e-)'3 (ni, n2) ^ \\fn, - fn,\\ < e; but by {/„ } ^''''^'' 
 and the fact that/ 4= z * ) * 1 1 / 1 1 = 1, we have ni 4= ^2 ' ) ' 1 1 /jh — /nj 1 1 = 1, 
 and these contradict each other when e < 1. Hence the result. 
 
 Thus the theorem is completely proved. 
 
 Corollary: Since every finite field is of the type* [p^[], where p is prime 
 and q a positive integer, we have : 
 
 3.5. Subfield R(^, x) of 365. — Since rational integral functions of a 
 transcendental x with coefficients in % form a subclass of 36j, viz., the class 
 of all finitely non-zero functions on ^ to 5, with z as the functional value 
 for negative values of the argument, the field of all rational functions of such 
 an ir is a subfield of 365. Thus, 
 
 Thm 3.6 %')• %, D R(5, x). 
 
 Further,! 
 
 ThmS.7 ^«°"« °' °' '""^ *yp« ^^ • ) • 3^5 3 R(^, a:) . 
 
 Proof: The proof is partly suggested by Hensel's proof of the theorem 
 that the field of the p-adic numbers includes properly the field of ordinary 
 rationals.J In analogy with Hensel we proceed to show that under the 
 conditions in the hypotheses every element of R(i5, x) corresponds to a 
 
 * M, p. 220. 
 
 t Here the symbol U reads "properly includes." 
 
 t H, p. 38. 
 
Gokhale: Concerning Compact Kiirschdk Fields. 313 
 
 periodic element of 3£g. A periodic element is defined thus: 
 
 D3.6 {h=ZM'^'"'^'":=:h:^:3(s,t>0)^i> s-)'fi= U^t, 
 
 the smallest such t is called the period. 
 
 All polynomials are obviously periodic, the period being 1 and the 
 repeated element 2. The product of two periodic elements is easily seen 
 to be periodic. Thus it is sufficient to prove that the reciprocal of a poly- 
 nomial ^ irreducible in i^ is a periodic element. 
 
 In case % is of the type "iP^, that is, the absolute algebraic field * charac- 
 teristic p, ip is of the first degree in x. In case (p = fx, f being a non-zero 
 element of ^, the reciprocal of (p is fx~'^, where / is the reciprocal of /, 
 and is, therefore, obviously periodic. In case (p = fx -\- fi where none of 
 /, /i are zero, its reciprocal Is /J/(l -\-f2x), where /J is the reciprocal of/i 
 and/ = /]/2. But 1/(1 + f2x) equals: 
 
 l-f2X+f2X'- ••• + (- iyf2X'-^ ••• 
 
 and since /2 is algebraic with respect to the field [0, 1, 2, • • - , p — l"] there 
 exists t a positive integer n such that/^ = 1. Hence the series is periodic. 
 In case % is finite, ^ is a Galois Field J [_p^']. If the polynomial <p 
 irreducible in ^ is of degree n in x, the class of all polynomials V' in a: with 
 coefficients in % falls modulo (p into p^^ classes of congruent polynomials, 
 and hence there exists § a positive integer m such that a;*" = 1 (mod. <p), 
 that is to say, there exists a polynomial xp such that <p(x)4/{x) = x^ — 1. 
 Thus: 
 
 lf<p(x) = xPix)llipix)4^(x)'] = ^{x)l{x-^ - 1) = - ■i/'(x) { 1 + a;- + 0:2- + . . . } 
 
 and is therefore obviously periodic. 
 
 3.6. Adjunction of Elements to X,^ algebraic to }^. — We shall now prove 
 a theorem required for the sequel : 
 
 where *S denotes a system of elements. 
 
 Proof: We put the elements of the two fields into isomorphism in this 
 manner: put the systems % and S in one into correspondence with those in 
 the other; since aS is algebraic with respect to '^, this correspondence will 
 persist under all the field operations like additions, etc. Also since the 
 moduli of these elements in % and S are the same in both cases the iso- 
 
 * Cf. 1.2; also S, p. 199. 
 t S., p. 251. 
 J M, p. 220. 
 
 § This can be proved by the usual methods of Galois Field theory (cf. L. E. Dickson, 
 "History of the Theory of Numbers," Washington, D. C, 1919). 
 
314 
 
 GoKBLA-LE : Concerning Compact Kiirschdk Fields. 
 
 morphism so far is complete. Also both 36,;<s) and 36,^(>S) are extensions of 
 ^(/S). If we now identify the element x and its moduli iti the two fields, 
 we see that both 36j<s) and ^^(S) are extensions of 36,^ Now 36j(»S) is the 
 smallest extension* of jEj containing the system S. Therefore 36,^s) o BEjCS). 
 Further every element of ^^^g) is of the form J^iSiX*, where the elements s 
 belong to \^{S), and is thus an element of %yiS). Thus the theorem is 
 completely proved. 
 
 IV. The Properties p, A, P, opt. 
 
 4.1. A Complete Existential Theory. — In the preceding sections we have 
 defined, with reference to Kiirschak fields, the four properties p, A, P, 
 and cpt. If the existence or non-existence of these properties in this order 
 is denoted by positive or negative signs in that order, a given Kiirschak 
 field will have one of the 2^= 16 characters (H — | — [- + ), (+H — I — ). 
 (H — I h), ••', i )• In this section we develop a complete exis- 
 tential theory^ of these properties, that is, we prove 2^ propositions stating 
 
 for each of the 2* combinations (+ + + + ), ••••> ( ) of these 
 
 properties whether there exists or does not exist a Kiirschak field having 
 the particular combination of the properties. These propositions are 
 tabulated below: 
 
 Thm 4.1-4.16: 
 
 Cases. 
 
 P- 
 
 A. 
 
 .P. 
 
 cpt. 
 
 Consistent. 
 
 2'.'.'.'.'.'.'.'.'.'.'. 
 
 3 
 
 4. 
 
 5 . 
 
 1 1 1 1 1 1 1 1 ++++++++ 
 
 1 1 1 1 ++++ 1 1 1 1 ++ + + 
 
 ++ 1 1 ++ 1 1 ++ 1 1 ++ 1 1 
 
 1 + I + I + I + I + I + I + I + 
 
 f + 1 + 1 
 
 6 
 
 + 
 
 7 
 
 8... 
 
 + 
 
 9 
 
 + 
 
 10 
 
 + 
 
 11 
 
 
 12 
 
 + 
 
 13 
 
 + 
 
 14 
 
 + - 
 
 15 
 
 
 16 
 
 + 
 
 
 
 Note: The + and — entries in the last column denote the existence and' 
 non-existence respectively of a Kiirschak field with the character denoted 
 by the entries under the first four columns. 
 
 Also in looking the entries under p, it is necessary to remember that the 
 entries in the final column do not depend upon any special choice of the 
 
 * S, p. 186. 
 t E. H. Moore. 
 Press, p. 82. 
 
 ' Introduction to a Form of General Analysis," 1910, Yale University 
 
Gokhale: Concerning Compact Kiirschdk Fields. 315 
 
 prime p. Thus the entries with + sign under p we interpret as p * ) ' *3 
 ^^•^^', and those with - sign as =^3 ^i"-"*"'. 
 
 4.2. Proofs: 
 
 (a) 3, 7, 11, and 15 follow from theorem 2.1, viz., ^''p* * ) * 9?^. 
 (6) 1 follows from theorem 2.4, viz., 9=^^'°^* ' ) ' ^°. 
 
 (c) The following number systems in analysis with the absolute value 
 as modulus prove 9, 12, 13, and 16 respectively: 
 
 (9) complex numbers, 
 
 (12) algebraic numbers, 
 
 (13) real numbers, 
 (16) rational numbers. 
 
 (d) To prove 2 : consider (X,^^)(Apy, this has the properties p, A, P. 
 By theorem 3.4, 36^p is P and — cpi. Using 2.15 we see that (X>^)(4p) 
 is — cpt 
 
 (e) To prove 5: consider 36 [pj. This field is ~A, for the equation 
 y^ — X = has no solution. However by 3.4 it is cpt. 
 
 ( / ) We get 6, an example being 2i^p. The proof is similar to the above. 
 (g) Similarly 36r where R = |^all ordinary rationals] gives us 14. 
 
 4.3. Proof of 10: We proceed to prove that: 
 
 Thm 4.IO0 ©up^ 
 
 where <S is the field of Hensel's p-adic numbers. 
 
 Take an element of ® whose modulus a is > 1. Take an infinite 
 sequence of positive distinct ordinary rationals {tn] so that they all lie 
 between two positive rationals r, r' . Let rn = Inffnn, where In, fUn are 
 positive and relatively prime. Consider the sequence {hn} where 
 hy - Afn = (n). Then || hn \\ = a^» (n) .'. {hn}^'''^'\ This sequence 
 an upper bound of whose modulus is gr(a^, a^'), cannot have any Cauchy 
 subsequence. 
 
 For otherwise, 
 
 e :) :3 ne ^{nu ni) > ne ' ) ' \\ K, — K^W < e, 
 but we have 
 
 ni + n2 : ) : II K, \\ + || K, || : ) : || hn, - hn, \\ = gr{\\ hn, \\, \\ hn, ||) 
 
 > 
 
 and taking e < a"", we get a contradiction. . 
 
 Hence the theorem. 
 
 4.4. Proof of 4: Consider OC[p])a; 36[p] is perfect by 3.3. If we now 
 prove that algebraic closure is in this case obtained by the adjunction of 
 an infinite number of elements, then, by 2.21 (3iip{)'2^ and hence by 2.1 
 
316 Gokbale: Concerning Compact Kiirschdk Fields. 
 
 — cpt. Now (X[p])a contains 3t.^ which by theorem 3.8 can be regarded 
 as obtained from X[p] by adjoining algebraic elements to [p]. The number 
 of these is, however, infinite ; otherwise ^^ would have only a finite number 
 of elements. Thus {2cip])a is infinite with respect to 96[pj and so the theorem 
 is proved. 
 
 4.5. Proof of 8: Ostrowski in his theses* has proved the following 
 theorem : 
 Thm 4.8o di'' ' »S«'«' ^= '"""^^ • ) ' ^(S) -^. 
 
 We use this theorem to prove 4.8. 
 
 Consider 36[p], p > 2; let r be any one of the integers: 2, 3, • • •, p — 1. 
 From Steinitz's paper, f we have 
 
 ^prlme. > p • \ . (yq A^ ~\ Irreducible In [p]. Ist kind 
 
 Consider a sequence of primes, {g„} such that 
 
 (1) qi>P 
 
 (2) Qn > p^i^^^s - «"-! 
 
 and the corresponding infinite system of irreducible polynomials 
 
 S^ (y^n-^r\n). 
 
 Consider now j6[p](<S). By theorem 3.8, 3lt[p](/S) = 3£[p](s). Also S is of 
 the first kind, further if Sm ^ (g«» ■^r\n= 1, 2, • • •, m - 1), MC^J 
 contains elements whose order is at most p^i'z ••• 9m-i and hence the poly- 
 nomial y^vi-\- r is irreducible in \jp^{Sm) since the order of its roots is 
 qM> the highest order in \jpJi{Sn,)- Thus 5 is a progressive J system. We 
 have further 'X,[p]{S)~^, since the polynomial y"^ -{- x for instance has' still 
 no root. Thus X,,] (5)^-^-^-"^*. 
 
 * Oi, p. 280. 
 1 8, p. 231. 
 
 tS, p. 271. 
 
VITA. 
 
 I, Vishnu Dattatreya Gokhale, was born in Poona City, India, on the 
 twenty-fourth of February, 1892. I got my secondary and high school 
 education at the Nutan Marathi Vidyalaya, Poona City. I entered 
 Fergusson College, Poona City, in 1907, and did all my undergraduate 
 and part of my graduate work there. My instructors in mathematics 
 were Prin. R. P. Paranjpye and Professor V. B. Naik. I got the degree of 
 Bachelor of Arts in 1911, and Master of Arts 1913, at the University of 
 Bombay. During 1914-1917 I worked as Professor of Mathematics at the 
 Fergusson College, Poona City. 
 
 I came to the United States in December, 1917. During 1918-19 I 
 took courses at the University of California under Professor Cajori, and 
 Drs. McDonald, Buck, Sperry and Bernstein. I joined the University of 
 Chicago in autumn, 1919, and during nine quarters took courses in mathe- 
 matics under Professors Moore, Bliss, Dickson and Wilczynski, the major 
 part of my work being under Professor Moore. 
 
 This paper was written under the direction of Professor Moore and I 
 wish to express my appreciation of his constant encouragement and in- 
 valuable criticism while engaged in the investigation. 
 
 I wish to acknowledge my indebtedness to all the men under whom I 
 have done my graduate work, I must mention Professors Paranjpye and 
 Naik, who created in me an interest in mathematics, and Dr. Sperry, who 
 advised and inspired me to go to Chicago. I cannot adequately express 
 what I owe to Professor Moore. While attending his lectures, and still 
 more in personal interviews, I enjoyed some of the best moments in my life 
 and it is he, more than any one else, who initiated me into the poetry of 
 mathematics and made me realize so vividly that mathematics is an art. 
 
52n;?9- 
 
 UNIVERSITY OF CAIIFORNU UBRARY