On Galois cohomology and realizability of 2-groups as Galois groups II

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2 ⁿ , n ≥ 4, having a cyclic subgroup of order 2 ⁿ⁻², over fields containing a primitive 2 ⁿ⁻³th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.     

Descent principle in modular Galois theory

We propound a descent principle by which previously constructed equations over GF(q _n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive)q-polynomial ofq-degreem with Galois group GL(m, q) and then, under suitable conditions, enlarging its Galois group to GL(m, q _n) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degreen. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.     

On Restricted Leibniz Algebras

ABSTRACT

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ?/n? with n = 4 or n = p ^r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.     

Automorphic lifts of prescribed types

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

0,1 distribution in the highest level sequences of primitive sequences over<Emphasis Type="Italic">Z</Emphasis><Subscript>2e</Subscript>

In this paper, we discuss the 0,1 distribution in the highest level sequence a_e-1 of primitive sequence over Z_2e generated by a primitive polynomial of degreen. First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight fore relatively small ton. We also get an estimate which is suitable fore relatively large ton. Combining the two bounds, we obtain an estimate depending only onn, which shows that the largern is, the closer to 1/2 the proportion of 1 will be.     

The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

Kummer extensions of rings

The theory of Kummer extensions of commutative rings is constructed, generalizing the theory of Kummer extension fields. A Galois extension S/R with Abelian Galois group of degreen is called Kummerian if in the group of invertible elements of the ringR there is a cyclic subgroup of ordern such that for any, 1 element 1- is invertible inR. Any cyclic Kummerian extension can be obtained by means of a suitable factorization of the tensor algebra over a finitely generated protectiveR -module of rank 1 (an arbitrary Kummerian extension is a tensor product of cyclic ones). The group of equivalence classes of Kummerian extensions with fixed Galois group is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 8–30, 1976.     

New classes of parameterized differential Galois groups

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups.

     

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, the ring of integers in the pth cyclotomic field, C _{f, p} : y ^p  =  f(x) the corresponding superelliptic curve and J(C _{f, p} ) its jacobian. Assuming that either n  =  p + 1 or p does not divide n(n  −  1), we prove that the ring of all endomorphisms of J(C _{f, p} ) coincides with . The same is true if n  =  4, the Galois group of f(x) is the full symmetric group S ₄ and K contains a primitive pth root of unity. An erratum to this article can be found at     

Algebraic function fields of one variable over finite fields are stable

It is shown that any algebraic curveC over a finite field has a separable cover of some degreen over the projective lineP ₁ such that the geometric Galois group of the Galois hull ofC |P ₁ is the full symmetric groupS _n. This work was partially supported by a grant from G.I.F. (German Israeli Foundation for Scientific Research and Development).     

A necessary and sufficient condition for a class of nonsingular Galois NFSRs

Let n be a positive integer. An n-stage Galois NFSR is nonsingular if and only if the output sequences of each bit register are purely periodic. It is well known that a Galois NFSR used to build a stream cipher should be nonsingular. Recently, a useful concept that is the simplified feedback function is proposed for a Galois NFSR, which is a vectorial Boolean function. Generally, for a Galois NFSR, its simplified feedback function has less nonlinear terms than its traditional feedback function, and so is easier to analyze. Moreover, it has been shown that the nonsingularity of a Galois NFSR is decided by the invertibility of its simplified feedback function. Based on this observation, in this paper, we present a necessary and sufficient condition for the nonsingularity of a class of Galois NFSRs such that every component of its simplified feedback is linear or has a common nonlinear term.     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

On the stable derivation algebra associated with some braid groups

We shall prove some stability property of the graded Lie algebraD _n of certain derivations associated with pure sphere braid group onn strings; in fact, thatD _n forn≥6. These Lie algebrasD _n are connected with some bigl-adic Galois representations, and the stability property is related to some conjecture of Grothendieck.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

The Invariant Subrings of an Azumaya Galois Extension

Let B be an Azumaya Galois extension or a DeMeyer-Kanzaki Galois extension with Galois group G. Equivalent conditions are given for a separable subextension of a Galois extension in the skew group ring B * G being an invariant subring of a subgroup of the Galois group G.AMS Subject Classification (2000): 16S35, 16W20.