The group [G.sub.P] = Gal(C(C)/[K.sub.P]) is called the

Galois group at P.

The main result of this theory merely says that if a Hamiltonian system is completely integrable, then the identity component of the

Galois group of the variational equation along certain particular solution is abelian.

If K [??] E is a finite Galois field extension and G is its

Galois group, then the opposite category of K-algebras A with E [[cross product].sub.K] A [approximately equal to] [E.sup.n] for some natural n is equivalent to the category of finite G-sets.

Conversely, every irreducible parabolic subgroup of rank k - 1 in [A.sub.n] is the

Galois group of some subspace in the k-equal arrangement.

The aim of this paper is to study the

Galois group of a certain factor of a 4-th dynatomic polynomial.

First of all, [S.sub.n] is canonically a profinite group, and Gal [equivalent to] Gal([F.sub.[p.sup.n]]/[F.sub.p]), the

Galois group of automorphisms of [F.sub.[p.sup.n]], acts continuously in an evident way on [S.sub.n].

then the

Galois group [G.sub.P] of the splitting field of P(x) over Q is of order at most 132.

On this basis rests the subsequent discussion of polynomials and their

Galois groups and representations of the

Galois group, followed by the reciprocity laws, including the famous proof by Andrew Wiles of Fermat's Last Theorem.

All of the examples are division algebra representatives of algebras of the form A [cross product] [delta]([chi]), where A is a k-division algebra and [delta]([chi]) is the cyclic k(t) or k((t))-division algebra defined by a character [chi] of the absolute

Galois group of k.