scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

scholarly journals Crossed Products and Ramification

1966 ◽  
Vol 28 ◽  
pp. 85-111 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Rank One ◽  
Definition Of

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

scholarly journals Ramification Theory for Extensions of Degree p

1971 ◽  
Vol 41 ◽  
pp. 149-168 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Residue Class ◽  
Field Extension ◽  
Quotient Field ◽  
Class Field ◽  
Wild Ramification ◽  
Valuation Rings ◽  
Ramification Theory ◽  
Residue Class Field ◽  
Rank One

The notions of tame and wild ramification lead us to make the following definition.Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

scholarly journals II-Principal Hereditary Orders

1968 ◽  
Vol 32 ◽  
pp. 41-65 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Extension ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Quotient Field ◽  
Group Of Units ◽  
Rank One ◽  
Hereditary Orders

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

scholarly journals Crossed Products and Hereditary Orders

1963 ◽  
Vol 23 ◽  
pp. 103-120 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Multiplicative Group ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Group Of Units ◽  
Rank One ◽  
Hereditary Orders

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).

scholarly journals Addendum: Π-principal hereditary orders (Nagoya Math. J. 32 (1968), 41–65)

1974 ◽  
Vol 54 ◽  
pp. 215-216
Author(s):  
Susan Williamson
Keyword(s):  
Direct Product ◽  
Galois Extension ◽  
Valuation Ring ◽  
Residue Class ◽  
Integral Closure ◽  
Quotient Field ◽  
Inertia Group ◽  
Residue Class Field ◽  
Rank One ◽  
Hereditary Orders

Let R denote a complete discrete rank one valuation ring of unequal characteristic, and let p denote the characteristic of the residue class field R̅ of R. Consider the integral closure S of R in a finite Galois extension K of the quotient field k of R. Recall (see Prop. 1.1 of [3]) that the inertia group G0 of K over k is a semi-direct product G0 = J × Gp, where J is a cyclic group of order relatively prime to p and Gp is a normal p-subgroup of G.

scholarly journals Equivalence Classes of Maximal Orders

1968 ◽  
Vol 31 ◽  
pp. 131-171 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Equivalence Classes ◽  
Brauer Group ◽  
Crossed Products ◽  
Quotient Field ◽  
Maximal Orders ◽  
Rank One

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

Defining Families for Integral Domains of Real Finite Character

1972 ◽  
Vol 24 (6) ◽  
pp. 1170-1177 ◽  
Author(s):  
William Heinzer ◽  
Jack Ohm
Keyword(s):  
Zero Element ◽  
Finiteness Condition ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Rank One

Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).

scholarly journals On Krull’s Conjecture concerning Valuation Rings

1952 ◽  
Vol 4 ◽  
pp. 29-33 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Valuation Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quotient Field ◽  
Counter Example ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Necessary And Sufficient

Previously W. Krull conjectured that every completely integrally closed primary domain of integrity is a valuation ring, The main purpose of the present paper is to construct in §1 a counter example against this conjecture. In § 2 we show a necessary and sufficient condition that a field is a quotient field of a suitable completely integrally closed primary domain of integrity which is not a valuation ring.

Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

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