On successful completion of this module, students will be able to
Justify with reasoned logical argument basic properties
of polynomial rings and finite field extensions.
Specify and justify with reasoned logical argument aspects
of the relationships between polynomials with coefficients
in some ground field, finite extensions of that ground field,
and the groups of automorphisms of those field extensions.
Determine the Galois groups of appropriately-chosen polynomials
of low degree.
The module will cover the following topics:
Basic principles of group theory.
Basic principles of ring theory.
Basic properties of polynomial rings with coefficients
in a field. Gauss's Lemma concerning products of
primitive polynomials. Eisenstein's irreducibility criterion
Basic properties of field extensions.
The Tower Law.
Basic properties of algebraic extensions. Proof that the
degree of a simple algebraic extension is equal to the degree
of the minimum polynomial of the adjoined element generating
Solvability and Insolvability of Ruler and Compass
Splitting fields. Existence and isomorphism theorems
for splitting fields. Normal extensions. Separability.
Basic properties of finite fields. The Primitive Element Theorem.
The Galois Group of a finite Field Extension.
The Galois Correspondence.
Procedures for determining the roots of quadratic, cubic
and quartic polynomials from the coefficients of such polynomials.
Galois groups of polynomials of low degree.
The class equation of a finite group. Cauchy's theorem
concerning the existence of elements of prime order in
finite groups. Simple groups. Solvable groups.
Galois's Theorem concerning the solvability of
polynomial equations. A quintic polynomial that is not
solvable by radicals.