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.     

The tame infinitesimal groups of odd characteristic

Given an infinitesimal group G over an algebraically closed field k of characteristic p?3, we provide criteria for the principal block B₀(G) of its algebra of distributions to be of tame representation type. These are employed in conjunction with Galois coverings to determine the structure of G modulo its multiplicative center as well as the quiver and the relations of the algebra B₀(G).     

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.     

On tame pro-p Galois groups over basic {\mathbb{Z}_p}-extensions

For a prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z_p}$ -extension of the rational number field. In this paper, we give a family of S for which the Galois group is a metacyclic pro-p group with an application to Greenberg’s conjecture.     

Equations with absolute Galois group isomorphic to An

Criteria are given for polynomials of the type Xⁿ + aX³ + bX² + cX + d, to have Galois group over any finite number field isomorphic to A_n. We use them to construct, for every n, infinitely many polynomials with absolute Galois group isomorphic to A_n, covering so, the case n even, $\text{4 ? n}$, for which explicit equations were not known.     

Artin's L-functions and one-dimensional characters

Let K/Q be a finite Galois extension with the Galois group G, let χ₁,…,χr be the irreducible non-trivial characters of G, and let A be the C-algebra generated by the Artin L-functions L(s,χ₁),…,L(s,χr). Let B be the subalgebra of A generated by the L-functions corresponding to induced characters of non-trivial one-dimensional characters of subgroups of G. We prove: (1) B is of Krull dimension r and has the same quotient field as A; (2) B=A iff G is M-group; (3) the integral closure of B in A equals A iff G is quasi-M-group.     

Covers of Semigroups and Function Semirings

We define and use a Galois correspondence to determine, for certain medial semigroups, S, some maximal semirings in the collection M(S) of self maps on S.     

On 3-class groups of cyclic cubic extensions of certain number fields

Let F be a (finite) algebraic number field, and let K be a cyclic cubic extension of F. Assuming that the 3-class group of F is trivial, we determine the rank of the 3-class group S_K of K, and we obtain some additional information about the structure of S_K.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.     

Galois number fields with small root discriminant

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω≈44.7632. We definitively treat the cases G=A₄, A₅, A₆ and S₄, S₅, S₆, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL₃(2), A₇, S₇, PGL₂(7), SL₂(8), ΣL₂(8), PGL₂(9), PΓL₂(9), PSL₂(11), and , and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G,Ω) is empty.     

Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

Circular units in a bicyclic field

For a real abelian field with a non-cyclic Galois group of order l², l being an odd prime, the index of the Sinnott group of circular units is computed.     

Polynomials with Dp as Galois group

A useful criterion characterizing a monic irreducible polynomial over $\text{Q}$ with Galois group D_p (the dihedral group of order 2p, p: prime) is given by making use of the geometry of D_p, i.e., D_p is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D₅ and D₇.     

Local-global principles for algebraic covers

This paper is devoted to some local-global type questions about fields of definition of algebraic covers. Letf:X→B be a covera priori defined over . Assume that the coverf can be defined over each completion ?{p} of ?. Does it follow that the cover can be defined over ?? This is thelocal-to-global principle. It was shown to hold for G-covers [DbDo], i.e., for Galois covers given with their automorphisms. Here we prove that, in the situation ofmere covers, the local-to-global principle holds under some additional assumptions on the groupG of the cover and the monodromy representationG→S _d (withd=deg(f)). This local-to-global problem is closely related to the obstruction to the field of moduli being a field of definition. This problem was studied in [DbDo], which is the main tool of the present paper.     

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.     

Deformations of Bloch groups and Aomoto dilogarithms in characteristic p

In this paper, we study the Bloch group B₂(F₂[ε]) over the ring of dual numbers of the algebraic closure of the field with p elements, for a prime p?5. We show that a slight modification of Kontsevich?s -logarithm defines a function on B₂(F₂[ε]). Using this function and the characteristic p version of the additive dilogarithm function that we previously defined, we determine the structure of the infinitesimal part of B₂(F₂[ε]) completely. This enables us to define invariants on the group of deformations of Aomoto dilogarithms and determine its structure. This final result might be viewed as the analog of Hilbert?s third problem in characteristic p.     

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π ₁(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π ₁(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π ₁ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.     

On Galois spectra of polynomials with integral parameters

We prove that there exists a polynomial F(x, t) with rational coefficients, whose degree with respect to x is equal to 4, such that for every integer a, the Galois group of the decomposition field of the polynomial F(x, a) is not the dihedral group, but any other transitive subgroup of the group S₄ can be represented as the Galois group of the decomposition field of the polynomial F(x, a) for a certain integer a. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 275–280.