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 On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

scholarly journals On weak center Galois extensions of rings

2001 ◽  
Vol 25 (7) ◽  
pp. 489-495
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Galois Extension ◽  
Galois Extensions ◽  
Weak Center

LetBbe a ring with 1,Cthe center ofB,Ga finite automorphism group ofB, andBGthe set of elements inBfixed under each element inG. Then, the notion of a center Galois extension ofBGwith Galois groupG(i.e.,Cis a Galois algebra overCGwith Galois groupG|C≅G) is generalized to a weak center Galois extension with groupG, whereBis called a weak center Galois extension with groupGifBIi=Beifor some idempotent inCandIi={c−gi(c)|c∈C}for eachgi≠1inG. It is shown thatBis a weak center Galois extension with groupGif and only if for eachgi≠1inGthere exists an idempotenteiinCand{bkei∈Bei;ckei∈Cei,k=1,2,...,m}such that∑k=1mbkeigi(ckei)=δ1,gieiandgirestricted toC(1−ei)is an identity, and a structure of a weak center Galois extension with groupGis also given.

scholarly journals On Azumaya Galois extensions and skew group rings

1999 ◽  
Vol 22 (1) ◽  
pp. 91-95
Author(s):  
George Szeto
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Group Rings ◽  
Galois Extensions ◽  
Skew Group Rings ◽  
Skew Group Ring

Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

scholarly journals On characterizations of a center Galois extension

2000 ◽  
Vol 23 (11) ◽  
pp. 753-758 ◽  
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Galois Extension ◽  
The Ideal

LetBbe a ring with1,  Cthe center ofB,  Ga finite automorphism group ofB, andBGthe set of elements inBfixed under each element inG. Then, it is shown thatBis a center Galois extension ofBG(that is,Cis a Galois algebra overCGwith Galois groupG|C≅G) if and only if the ideal ofBgenerated by{c−g(c)|c∈C}isBfor eachg≠1inG. This generalizes the well known characterization of a commutative Galois extensionCthatCis a Galois extension ofCGwith Galois groupGif and only if the ideal generated by{c−g(c)|c∈C}isCfor eachg≠1inG. Some more characterizations of a center Galois extensionBare also given.

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 Mahler measure on Galois extensions

2018 ◽  
Vol 14 (06) ◽  
pp. 1605-1617 ◽  
Author(s):  
Francesco Amoroso
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Extension ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Galois Extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

scholarly journals The Galois extensions induced by idempotents in a Galois algebra

2002 ◽  
Vol 29 (7) ◽  
pp. 375-380
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Central Idempotent ◽  
Galois Extensions ◽  
Direct Sum ◽  
Equivalence Condition ◽  
Minimal Idempotent

LetBbe a Galois algebra with Galois groupG,Jg={b∈B|bx=g(x)b   for all   x∈B}for eachg∈G,egthe central idempotent such thatBJg=Beg, andeK=∑g∈K,eg≠1egfor a subgroupKofG. ThenBeKis a Galois extension with the Galois groupG(eK)(={g∈G|g(eK)=eK})containingKand the normalizerN(K)ofKinG. An equivalence condition is also given forG(eK)=N(K), andBeGis shown to be a direct sum of allBeigenerated by a minimal idempotentei. Moreover, a characterization for a Galois extensionBis shown in terms of the Galois extensionBeGandB(1−eG).

The structure of a partial Galois extension II

2016 ◽  
Vol 15 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Finite Group ◽  
Galois Extension ◽  
Structure Theorem ◽  
Galois Extensions ◽  
Partial Action ◽  
A Finite Group ◽  
Partial Galois Extension

Let [Formula: see text] be a partial Galois extension where [Formula: see text] is a partial action of a finite group on a ring [Formula: see text] such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in [Formula: see text], and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for [Formula: see text].

Cyclotomic Galois module structure and the second Chinburg invariant

1995 ◽  
Vol 117 (1) ◽  
pp. 57-82
Author(s):  
Victor P. Snaith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Prime Power ◽  
Class Group ◽  
Number Fields ◽  
Affirmative Answer ◽  
Module Structure ◽  
Integral Group ◽  
Galois Module

AbstractWe study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

scholarly journals A non-abelian Stickelberger theorem

2010 ◽  
Vol 147 (1) ◽  
pp. 35-55 ◽  
Author(s):  
David Burns ◽  
Henri Johnston
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Class Group ◽  
Number Fields ◽  
Stickelberger Theorem ◽  
Use Values

AbstractLet L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

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