On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Let T be an integral domain with a maximal ideal M, ?: T → K: = T/M the natural surjection, and R the pullback ?^?1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x ₁]]…[x _n ]] to be catenarian, where each [x _i ]] is fixed as either [x _i ] or [[x _i ]]. We also give a complete answer to the question of determining the field extensions k ? K such that the contraction map Spec(K[x ₁]]…[x _n ]]) → Spec(k[x ₁]]…[x _n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x ₁]]…[x _n ]] is catenarian.     

Class number one quadratic fields and solvability of some Pellian equations

Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that q≡r≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) p∈N satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z ²-dY ² = ±4^δ p is solvable in integers. (c) For every p∈П the Pellian equation W ²-dV ² = 4^δ p ² has a solution (w,v) in integers such that gcd (w,v) divides 2^δ.     

Generators for decompositions of tensor products of modules associated with standard Jordan partitions

If K is a field of finite characteristic p, G is a cyclic group of order q = p^α, U and W are indecomposable KG-modules with dim U = m and dim W = n, and λ(m,n,p) is a standard Jordan partition of mn, we describe how to find a generator for each of the indecomposable components of the KG-module U ? W.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

Some Maximal Function Fields and Additive Polynomials

We derive explicit equations for the maximal function fields F over 𝔽 _{q ²ⁿ} given by F = 𝔽 _{q ²ⁿ} (X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field 𝔽 _{q ²ⁿ} , and where A(Y) is q-additive and deg(f) = q ⁿ  + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over 𝔽 _{q ²ⁿ} (i.e., the extension H/F is Galois).     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

Tame Kernels of Quaternion Number Fields

Let E/F be a Galois extension of number fields with the quaternion Galois group Q ₈. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel of E with the same orders of some of its subfields. Let E/? be a Galois extension of number fields with the Galois group Q ₈ and p an odd prime such that p ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that the p-Sylow subgroup of the tame kernel is nontrivial, then p ^r -rank(K ₂(E/K)) is even, i.e., 2|p ^r -rank(K ₂(𝒪 _E )) ? p ^r -rank(K ₂(𝒪 _K )), where K is the quartic subfield of E.     

Moufang Loops of Odd Orderpq1 ··· qnr1 ··· rm

LetLbe a Moufang loop of odd orderpαqα11···qnαnwherepandqiare primes with 3 ≤ p < q1 < ··· < qnand αi ≤ 2. In this paper, we prove thatLis a group ifpandqiare primes with 3 ≤ p < q1 < ··· < qn: (i) α ≤ 3, or (ii) α ≤ 4,p ≥ 5.     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

Construction of somep-extensions of the rational number field

In this paper, we constructp-extensionsK _a ,a(modp ^r ), of degreep ^3r,p≠2, r>0, of the field ℚ of rational numbers with ramification pointsp andq. The Galois groupG(K _a )/ℚ of the extensionK _a /ℚ,a(modp ^r ), is defined by the generators and relations

The Lie Module Function for Symmetric Groups #

Let V be a ? G-module where ? is the field of all complex numbers and G is a symmetric group. The purpose of this article is to give a method of analyzing the Lie powers L ⁿ (V ), for every positive integer n, by making use of the recent work of Bryant.     

On Residually Finite Groups with Engel-like Conditions

Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) ≤ m such that x^q is left n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x^q is left n-Engel. Then w(G) is locally virtually nilpotent.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

Stability of Numerical Invariants in Free Groups

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l ₁ and l ₂ be cyclic field extensions of k of degree d with norms n _{l 1/k} and n _{l 2/k} . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n _{l 1/k}  ? n _{l 2/k} of degree d. Let K = k(X ₁,…, X _{d ²} ). We compute polynomials which are similarity factors of a form of the kind N ? (n _{l 2/k}  ? _k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n _{l 1/k}  ? n _{l 2/k} to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n _{l 1/k} .     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.