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Differents, Finite Bases, and Prime Decomposition in Almost-Dedekind Domains

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Differents, Finite Bases, and Prime Decomposition in Almost-Dedekind Domains Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1970 Differents, Finite Bases, and Prime Decomposition in Almost-Dedekind Domains. Robert William Yeagy Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Yeagy, Robert William, "Differents, Finite Bases, and Prime Decomposition in Almost-Dedekind Domains." (1970). LSU Historical Dissertations and Theses. 1898. https://digitalcommons.lsu.edu/gradschool_disstheses/1898 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. 71-6618 YEAGY, Robert William, 1943- ' DIFFERENTS, FINITE BASES, AND PRIME DECOMPOSITION IN ALMOST-DEDEKIND DOMAINS. The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1970 Mathematics University Microfilms, Inc., Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED DIFFERENTS, FINITE BASES, AND PRIME DECOMPOSITION IN ALMOST-DEDEKIND DOMAINS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Meehanicai College In partial fulfillment of the requirements for the degree of Doctor of Philosophy i n The Department of Mathematics by Robert W illi am Yeagy B.A,, Franklin and Marshall College, 196** M.S., Louisiana State University, 1966 August, 1970 ACKNOWLEDGMENT This dissertation was written under the supervision of Dr. Hubert S. Butts, Professor of Mathematics. The author is grateful to Dr. Butts for his teaching and counsel and for his help in the development of the dissertation. TABLE OF CONTENTS page Abstract - — ------------------------------------------------------------------------------- iv Chapter I Introduction----------------------------------------------- —---------------- 1 II Finite Bases ------------------------------------------------------------------ 8 III Free, Finite Bases ------------ 25 IV The Weil Different -- ---------------------------------------------------- 30 V Prime Decomposition ---------------------------------------------------- 37 References ------------------------------------------- — **k Vita .......................... - .............................................- .................................. ^6 ABSTRACT Let R be an almost-Dedekind domain - that is, suppose that R is an integral domain with 1 ^ 0 such that Rp is a rank-one discrete valuation ring for every maximal ideal P of R. Let K be the quotient field of R and let R' be the integral closure of R in a finite separable algebraic extension K'/K. The main topic of this disser­ tation is the existence of a fin ite basis for R1 over R. If d is the ideal of R' generated by the differents of all elements of R' and if S is the fractionary ideal of R1 generated by {£ eK1|T(a£) € R for all ae R1), then there exists such a fin ite basis if dS = R1. (For this result we assume only that R is an integrally closed domain with 1 £ 0). We show that R‘ is a fin ite R-module if R consists of algebraic integers or, in certain cases, if R is the almost-Dedekind domain constructed by Heinzer and Ohm. We also study conditions for the existence of a fin ite , linearly independent basis for R' over R, The ideal [R'lSj^, is called the different of R1 over R, If R is noetherian, various equivalent definitions for the different are known, including one proposed by Weil. The construction of Weil can be carried through if R is almost-Dedekind, and we do so. If R' is a fin ite R-module, the ideal so constructed is [R'lS]..,. I\ Let R be an almost-Dedekind domain which contains a zero £ 1 of xp - 1 where p is a prime natural number. Suppose |j ( R has no p-th root in K, and let R1 denote the integral closure of R in K(fi1/p). If P is a maximal ideal of R, we classify the factorizations of PR1, according to whether p^m or p|m and the congruences xp=/u(mod P-*) can be solved in R for certain natural numbers j . CHAPTER I INTRODUCTION The main objects of study in this dissertation are rank-one discrete valuation rings and almost-Dedekind domains. An integral domain R with 1 ^ 0 is called almost- Dedekind if Rp is a rank-one discrete valuation ring for every maximal ideal P of R. There are examples of such rings in [N] and [HOJ, and there is a chapter in [G] on almost-Dedekind domains. We will review some of their basic propert ies. Suppose that J is an integral domain with 1 ^ 0 which is a subring of a field F and that A is a J-submodule of F. If E is a multiplicative system in J, then we define the quotient module A^ as the set [am ^jacA.mcE}. If P is a prime ideal of J, we abbreviate aj V t0 V Lemma i : A = f| p where P runs through the maximal ideals of J. Proof; Evidently A cr n p Ap. Suppose that a e D p Ap. For every P, there exist 3p f A and nip e J\P such that a = Spin p \ Then (A:aJ)j lies in no maximal ideal of J, so (A :aj)j = J and a c A. Q., E, D. Of course all almost-Dedekind domains are one-dimensional and Prufer, and they have many properties of Dedekind domains. In particular if R is an almost-Dedekind domain and tf A is an integral ideal of R which lies in only fin ite ly many maximal ideals P p ..,P of R, then al a " A = Pj . ...P for some positive integers a.. This is easy to prove. For each maximal ideal P of R, the ideal ARp DR is P-primary. Since Rp is a rank-one discrete valuation ring, the only P-primary ideals of R are the powers of P [ZS, 153 & 225], Letting P run over all maximal 2 ideals of R, we have A = Hp (ARp n R) = 0p Pa^ . l f P | t ...,P afP ) are the only maximal ideals of R containing A, then A = f|.n, P. i . i = l i ’ a( P 1 and since these primary ideals are pair-wise comaximal, A=D.|]j P. i [ZS, 177]. Let R be an almost-Dedekind domain with quotient field K. We w ill study questions which arise in a fin ite separable algebraic extension K1 of K, where the integral closure R1 of R is always an almost-Dedekind domain [G, 29.1]. If P is a maximal ideal of R, then the integral closure of Rp in K1 is R’p [G, 8.2] and one can show that R'p is an almost-Dedekind domain by using the tran sitivity of quotient ring formation [ZS, 226]. However, R'p has only fin ite ly many maximal ideals since only fin ite ly many maximal ideals of R1 can 1ie over P in R [G, 10.5]. Therefore R'p is a P1D [ZS, 278]. Let P be a maximal ideal of the almost-Dedekind domain R and let M ,, . ,M be the maximal ideals of R1 which lie over P in R. Then 1 9 6 £1 PR1 = Mj 1 , .. 9 for some posi t ive integers e . , We wi 11 show that L j^ je .f. = (K1: K) where f. = (R'/M.: R/P) . This equality is well known if the ground ring is a Dedekind domain [ZSf 287]. So we go to Rp; we have (PRp)R'p = II.9, (M.R'p) 0 ' and f. = (R1p/(M.R1p) : Rp/PRp) (the M.R'p being maximal ideals of R'p), so e .f. = (K':K). If the extension K'/K is normal, then it is also known that e. = ... =* e (= e) and f = ... = f (=f) , and hence efg = (K1: K) I n I n ' [ZS, 289]. These facts are known as the "efg^theorem.11 A notion of some interest to us is the group of divis ib i1 t tv of an integral domain. Let J be an integral domain with 1 ^ 0 and quotient field F. If we let F" = F\{0} and denote by U(J) the the multiplicative group of units in J, then the factor group F*'/U(j) is called the group of d iv is ib ility of J. It is customary to write the operation in G as addition with identity element 0. We can i'c — introduce a p artial order into G in this way: if xeF then x ^ 0 if and only if xeJ. An interesting situation occurs if we assume that G is a la ttic e - i . e . , that every two elements of G have an infinum in G. One can then establish a one-to-one inclusion-preserving correspondence between proper ideals of J and segments of G, defined thus: a set S i* G of non-negative elements in G is called a segment provided (I) if aeS and beG, b & a, then beS ( i i ) If a, beS, then inf(a,b)e$ (see [G, App. *♦]). If J is an integral domain with quotient field F and if J' is the integral closure of J in a finite separable algebraic extension F'/F, then F1 ■ F(0) for some fleJ1. We know that F' = F (tf) for some rjfF1 [vdW 126], For some f(x)cj[x3, we have f(lj) =» 0; let c be the leading coefficient of f (x ) . I f 6 =» cij, then fleJ' and F1 = F(0) since 1)CF(0) • Let F be a fie ld and F1 a fin ite separable algebraic extension of F. I f F" is the normal closure of F '/F , then let ^ ,..., be coset representatives for the factor group Ga!(F":F)/ Gal(F1,:F').
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