Galois theory translates questions about fields into questions about groups. The fundamental theorem of Galois theory states that there is a bijection between the intermediate fields of a field extension and the subgroups of the corresponding Galois group. After a basic introduction to category and Galois theory, this project recasts the fundamental theorem of Galois theory using categorical la...

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±pA for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL2(5).10, and field discriminant 569. A closely related polynomial has degree 120, Galois group of the form SL2(5).20, and field discriminant 5311. We completely describe 5-adic behavior, f...

We consider finite Galois extensions of Qp and deduce bounds on the discriminant of such an extension based on the structure of its Galois group. We then apply these bounds to show that there are no Galois extensions of Q, unramified outside of {2,∞}, whose Galois group is one of various finite simple groups. The set of excluded finite simple groups includes several infinite families. Understan...

We construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 ove...

We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f , without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q[x]. For computing such polynomials, we use quadratic Newton-lifting and truncated expressions of the roots of f ove...

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...

We construct and study the moduli of continuous representations of a profinite group with integral p-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a p-adic local field, we show that these moduli spaces admit Zariski-closed loci cutti...

This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions t...

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z(p)[G] that annihilates the p-part of the class group of L.

Examples of polynomials with Galois group over Q(t) corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over Q through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.