Despite its name, the Fundamental Theorem of Algebra cannot be a result in pure algebra since the real numbers and hence the complex numbers are not algebraically defined. While there are many proofs, most use some basic facts in complex analysis or plane topology. We describe here a proof based on Galois theory as well as some non-trivial finite group theory, namely the Sylow theorems, but whi...

This is the material which I presented at the 60th birthday conference of my good friend José Luis Vicente in Seville in September 2001. It is based on the nine lectures, now called sections, which were given by me at Purdue in Spring 1997. This should provide a good calculational background for the Galois theory of vectorial (= additive) polynomials and their iterates.

Given an abelian algebraic group A over a global field F , α ∈ A(F ), and a prime `, the set of all preimages of α under some iterate of [`] generates an extension of F that contains all `-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as p...

Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by AutE. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of AutE consisting of all automorphisms σ : E → E such that σ(a) = a for all a ∈ F . We write this last c...

Let f(x) 2 Z[x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1=r1, r2, 1=r2, . . . , rd, 1=rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 +1=r1, . . . , rd +1=rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg = Gf=N , where N is isomorphic to a subgroup of Cd 2 . In a naive sense, the generic case is...

Clearly F ≤ EH ≤ E. On the other hand, given an intermediate field K between F and E, i.e. a subfield of E containing F , so that F ≤ K ≤ E, we can define Gal(E/K) and Gal(E/K) is clearly a subgroup of Gal(E/F ), since if σ(a) = a for all a ∈ K, then σ(a) = a for all a ∈ F . Thus we have two constructions: one associates an intermediate field to a subgroup of Gal(E/F ), and the other associates...

We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group” U∗, which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U. The group U arises through a Riemann–Hilbert correspondence....

We correct some errors in Grillet [2], Section V.9.

Galois is a domain specific language supported by the Galculator interactive proof-assistant prototype. Galculator uses an equational approach based on Galois connections with indirect equality as an additional inference rule. Galois allows for the specification of different theories in a point-free style by using fork algebras, an extension of relation algebras with expressive power of first-o...

4 Introduction to Galois Theory 2 4.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Gauss’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Eisenstein’s Irreducibility Criterion . . . . . . . . . . . . . . . 6 4.4 Field Extensions and the Tower Law . . . . . . . . . . . . . . 6 4.5 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . 8 4....