scholarly journals On the Quadratic Extensions and the Extended Witt Ring of a Commutative Ring

1973 ◽  
Vol 49 ◽  
pp. 127-141 ◽  
Author(s):  
Teruo Kanzaki
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Identity Element ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Witt Ring ◽  
Division Rings ◽  
Quadratic Extensions ◽  
The Common

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

scholarly journals On automorphism group of free quadratic extensions over a ring

1984 ◽  
Vol 7 (1) ◽  
pp. 103-108
Author(s):  
George Szeto
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Galois Extension ◽  
Zero Divisor ◽  
Quadratic Extension ◽  
Normal Extension ◽  
Quadratic Extensions ◽  
The One

LetRbe a ring with1,ρan automorphism ofRof order2. Then a normal extension of the free quadratic extensionR[x,ρ]with a basis{1,x}overRwith anR-automorphism groupGis characterized in terms of the element(x−(x)α)forαinG. It is also shown by a different method from the one given by Nagahara that the order ofGof a Galois extensionR[x,ρ]overRwith Galois groupGis a unit inR. When2is not a zero divisor, more properties ofR[x,ρ]are derived.

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 The Galois algebras and the Azumay Galois extensions

2002 ◽  
Vol 31 (1) ◽  
pp. 37-42
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Galois Extensions

LetBbe a Galois algebra over a commutative ringRwith Galois groupG,Cthe center ofB,K={g∈G|g(c)=c for all c∈C},Jg{b∈B|bx=g(x)b for all x∈B}for eachg∈K, andBK=(⊕∑g∈K Jg). ThenBKis a central weakly Galois algebra with Galois group induced byK. Moreover, an Azumaya Galois extensionBwith Galois groupKis characterized by usingBK.

scholarly journals Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

scholarly journals The rationality problem for norm one tori

2011 ◽  
Vol 202 ◽  
pp. 83-106
Author(s):  
Shizuo Endo
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Field Extension ◽  
Galois Closure ◽  
Rationality Problem

AbstractWe consider the problem of whether the norm one torus defined by a finite separable field extensionK/kis stably (or retract) rational overk. This has already been solved for the case whereK/kis a Galois extension. In this paper, we solve the problem for the case whereK/kis a non-Galois extension such that the Galois group of the Galois closure ofK/kis nilpotent or metacyclic.

On 2-Groups as Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1253-1273 ◽  
Author(s):  
Arne Ledet
Keyword(s):  
Galois Group ◽  
Direct Product ◽  
Galois Extension ◽  
Quaternion Algebra ◽  
Abelian Groups ◽  
Galois Groups ◽  
Embedding Problem ◽  
Brauer Group ◽  
Characteristic 2 ◽  
Cyclic Kernel

AbstractLet L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 08 (04) ◽  
pp. 493-503 ◽  
Author(s):  
H.-J. BARTELS ◽  
D. A. MALININ
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Extension ◽  
Commutator Subgroup ◽  
Finite Extension ◽  
Number Fields ◽  
Maximal Order ◽  
Linear Groups ◽  
Stable Linear

Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups [Formula: see text] which are not fixed elementwise by the commutator subgroup of Gal (K/R).

scholarly journals On central commutator Galois extensions of rings

2000 ◽  
Vol 24 (5) ◽  
pp. 289-294
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Structure Theorem ◽  
Projective Group ◽  
Galois Extensions ◽  
Separable Extensions

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

scholarly journals Computation of Hilbert sequence for composite quadratic extensions using different type of primes inQ

1997 ◽  
Vol 20 (1) ◽  
pp. 43-46
Author(s):  
M. Haghighi ◽  
J. Miller
Keyword(s):  
Galois Group ◽  
Quadratic Extensions ◽  
Definition Of

First, we will give all necessary definitions and theorems. Then the definition of a Hilbert sequence by using a Galois group is introduced. Then by using the Hilbert sequence, we will build tower fields for extensionK/k, whereK=k(d1,d2)andk=Qfor different primes inQ.

Simple reduction theorems for extension and torsion functors

1961 ◽  
Vol 57 (3) ◽  
pp. 483-488
Author(s):  
D. G. Northcott
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Residue Class ◽  
Identity Element ◽  
Negative Integer ◽  
Reduction Theorem ◽  
Residue Class Ring ◽  
Image Position ◽  
Class Ring ◽  
Simple Reduction

Let Λ be a commutative ring with an identity element and c an element of Λ which is not a zero divisor Denote by Ω the residue class ring Λ/Λc. If now M is a Λ-module for which c is not a zero divisor, and A is an Ω-module, then a theorem of Rees (2) asserts that, for every non-negative integer n, we have a Λ-isomorphismThis reduction theorem has found a number of useful and interesting applications.

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