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 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 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).

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 Integral Normal Bases in Galois Extensions of Local Fields

1970 ◽  
Vol 39 ◽  
pp. 141-148 ◽  
Author(s):  
S. Ullom
Keyword(s):  
Prime Ideal ◽  
Galois Group ◽  
Galois Extension ◽  
Residue Field ◽  
Local Fields ◽  
Galois Extensions ◽  
Normal Bases ◽  
Ring Of Integers

Throughout this paper F denotes a field complete with respect to a discrete valuation, kF the residue field of F, K/F a finite Galois extension with Galois group G = G(K/F). The ring of integers 0K of K contains the (unique) prime ideal ; the collection of ideals n for all integers n are ambiguous ideals i.e. G-modules.

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 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]).

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