The Galois group of a radical extension of the rationals

The Galois group of the splitting field of an irreducible binomialx ^2e −a overQ is computed explicitly as a full subgroup of the holomorph of the cyclic group of order 2 ^e . The general casex ⁿ −a is also effectively computed.     

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

In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.     

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.     

On mappings of the Galois space

LetV be a metric vector space over a fieldK, dimV=n<∞, and let δ:V×V→K denote the corresponding distance function. Given a mappingσ:V→V such that δ(p,q) = 1⇒ δ(p ^σ ,q ^ς) = 1, ifn=2, indV=1 and charK≠2, 3, 5, thenσ is semilinear [5], [11]; ifn≧3,K=R and the distance function is either Euclidean or Minkowskian, thenσ is linear [3], [10]. Here the following is proved: IfK=GF(p ^m ),p>2 andn≧3, thenσ is semilinear (up to a translation), providedn≠0, −1, −2 (modp) or the discriminant ofV satisfies a certain condition. The proof is based on the condition for a regular simplex to exist in a Galois space, which may be of interest for its own sake.     

Automatic realizability of Galois groups of order 16

This article examines the realizability of small groups of order , as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.

     

On unramified Galois extensions constructed using Galois representations

 For a field k, We denote the maximal abelian extension of k by k _ab and (K _ab ^r−1 _ab by k _ab ^r . In this paper, unramified Galois extensions over k _ab ^r are constructed using Galois representations of arbitrary dimension with larger coefficient rings. Received: 31 August 2001 / Revised version: 22 March 2002 Mathematics Subject Classification (2000): 11R21     

Small maximal pro-p Galois groups

For an odd prime p we classify the pro-p groups of rank ≤ 4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of order p. Received: 2 September 1997     

Quotients of absolute Galois groups which determine the entire Galois cohomology

For a prime power q = p ^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F,\mathbbZ/q){H^*(G_F,\mathbb{Z}/q)} is determined by a quotient G_F^[3]{G_F^{[3]}} of the absolute Galois group G _F related to its descending q-central sequence. Conversely, we show that G_F^[3]{G_F^{[3]}} is determined by the lower cohomology of G _F . This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.     

Galois realizability of groups of order 64

This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 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 the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.     

Harmonic Analysis on a Galois Field and Its Subfields

Complex functions χ(m) where m belongs to a Galois field GF(p ^ℓ ), are considered. Fourier transforms, displacements in the GF(p ^ℓ )×GF(p ^ℓ ) phase space and symplectic transforms of these functions are studied. It is shown that the formalism inherits many features from the theory of Galois fields. For example, Frobenius transformations and Galois groups are introduced in the present context. The relationship between harmonic analysis on GF(p ^ℓ ) and harmonic analysis on its subfields, is studied.      

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extensionN of a givenCM-fieldK are invariant under the involution on the set of all realizations of Δ overK which is induced by complex conjugation onK and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number fieldK are completely characterized by ramification and Galois module structure.     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Galois groups of intersections of local fields

LetK be a denumerable Hilbertian field with separable algebraic closure and Galois group , letw ₁,...w _n be absolute values on . Then for almost allσ ∈ G _K ⁿ (in the sense of Haar measure) there are no relations between the decomposition groups G _K (ω ₁ σ ₁),...,G _K (w _n σ _n ) of the absolute valuesw ₁ σ ₁,...,w _n σ _n i.e. the subgroup of G _K generated by these groups is the free product of these groups.     

Construction of some wreath products as Galois groups of normal real extensions of the rationals

Let K be a real algebraic number field. Suppose that G occurs as a Galois group of a normal real extension field of K. Using elementary methods, we show that certain types of split extensions of an elementary abelian 2-group by G also occur as Galois groups of normal real extensions of K. Among other examples, we show that Sylow 2-subgroups of the symmetric and alternating groups of degree 2ⁿ, as well as the Weyl groups of type B_n and D_n, occur as Galois groups of real extensions of the rationals.     

On cardinalities of row spaces of Boolean matrices

We prove that there are exactlyn numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2 ⁿ⁻¹+1,2 ⁿ⁻¹+2,2 ⁿ⁻¹+4, ..., 2 ⁿ⁻¹+2 ⁿ⁻², 2 ⁿ . Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2 ⁿ⁻¹+n−1. The second is that the numbers listed above are precisely the numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of topologies on a finite setX withn elements.     

Černý’s conjecture and group representation theory

Let us say that a Cayley graph Γ of a group G of order n is a Černy Cayley graph if every synchronizing automaton containing Γ as a subgraph with the same vertex set admits a synchronizing word of length at most (n−1)². In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of Černy Cayley graphs.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.