We describe methods for the computation of Galois groups of univariate polynomials over the rationals which we have implemented up to degree 15. These methods are based on Stauduhar's algorithm. All computations are done in unramiied p-adic extensions. For imprimitive groups we give an improvement using subbelds. In the primitive case we use known subgroups of the Galois group together with a c...

This article outlines techniques for computing the Galois group of a polynomial over the rationals, an important operation in computational algebraic number theory. In particular, the linear resolvent polynomial method of [6] will be described.

Galois corings with a group-like element [4] provide a neat framework to understand the analogies between several theories like the Faithfully Flat Descent for (noncommutative) ring extensions [26], Hopf-Galois algebra extensions [27], or noncommutative Galois algebra extensions [23, 15]. A Galois coring is isomorphic in a canonical way to the Sweedler’s canonical coring A ⊗B A associated to a ...

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...

1. Examples Example 1.1. The field extension Q(√ 2, √ 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are determined by their values on √ 2 and √ 3. The Q-conjugates of √ 2 and √ 3 are ± √ 2 and ± √ 3, so we get at most four possible automorphisms in the Galois group. See Table 1. Since the Galois group has order 4, these 4 possible assignments of v...

We prove: A proper profinite group structure G is projective if and only if G is the absolute Galois group structure of a proper field-valuation structure with block approximation. MR Classification: 12E30 Directory: \Jarden\Diary\HJPa 30 April, 2008 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. ** Research partially don...

We use elementary algebraic methods to reprove a theorem which was proved by Pop using rigid analytic geometry and in a less general form by Harbater using formal algebraic patching: Let C be an algebraically closed field of cardinality m. Consider a subset S of P(C) of cardinality m. Then the fundamental group of P(C) rS is isomorphic to the free profinite group of rank m. We also observe that...

In this paper we show that the polynomial x − 5x +12x − 28x +72x − 132x + 116x − 74x + 90x − 28x − 12x + 24x − 12x − 4x − 3x − 1 ∈ Q[x] has Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunther Malle (see [12]). The computation of this polynomial uses modular forms and their Galois representations.

Supported by the Graduiertenkolleg \Analyse und Konstruktion in der Mathematik".

We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK [G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extensio...