Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th roo...

This paper is an enlarged version of the lecture given at the AMS conference “Motives” in Seattle, July 1991. More details can be found in [G2]. My aim is to formulate a precise conjecture about the structure of the Galois group Gal (MT (F )) of the category MT (F ) of mixed Tate motivic sheaves over Spec F , where F is an arbitrary field. This conjecture implies (and in fact is equivalent to) ...

Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.

We describe methods for explicit computation of Galois groups of certain tamely ramified p-extensions. In the finite case this yields a short list of candidates for the Galois group. In the infinite case it produces a family or few families of likely candidates.

If the absolute Galois group GK of a field K is a direct product GK = G1 × G2 then one of the factors is prosolvable and either G1 and G2 have coprime order or K is henselian and the direct product decomposition reflects the ramification structure of GK . So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of ab...

It is well known that the Galois group of an extension L/F puts constraints on the structure of the relative ideal class group Cl(L/F ). Explicit results, however, hardly ever go beyond the semisimple abelian case, where L/F is abelian (in general cyclic) and where (L : F ) and #Cl(L/F ) are coprime. Using only basic parts of the theory of group representations, we give a unified approach to th...

The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be reg...

This paper surveys recent work on the problems of calculating Galois groups of diierentialequations and of constructingdiierentialequations with given groups as their Galois groups.

Arboreal Galois groups sit naturally as subgroups of tree (or graph) automorphism groups, while dynatomic Galois groups are naturally subgroups of certain wreath products. A fundamental problem is to determine general conditions under which these dynamically generated Galois groups have finite index in the natural geometric groups that contain them. This is a dynamical analog of Serre’s theorem...

(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. ...