Fields and their automorphisms
Reading: [Artin] pp. 493-508.
Concepts of algebraic, transcendental for elements, eld extensions; discuss. Finite degree implies
Let k be any eld and f (t) k[
Crash course in modules and presentation matrices
Material covered in Artin Chapter 12 section 1-6.
that most, but not necessarily all of our rings are commutative with unit, and were talking about
modules over such rings. Recall that all our rin
Let G be a nite group. Recall that E = EG is the complex vector space of (complex-valued) functions on the set of conjugacy classes of G endowed with the (nondegenerate) Hermitian conjugateinner-product) form
Abelian representations, and character groups
Let G be a group (any group). Let K be a eld. A one-dimensional representation r of the group
G over the eld K can be given in various equivalent ways:
1. by giving V a one-di
Duality, and Intro to Galois Theory
Duality in groups
Let G be a group and A an abelian group (lets write multiplicatively).
Denition 1 By Homgroups (G, A) = Hom(G, A) = Hom(Gab , A) we mean the abelian group of
homomorphisms from G to A where the multi
Gausss Lemma for Q:
Dene the content of a polynomial in Q[t]. Dene primitive polynomial ( Z[t]). Any (nonzero)
polynomial in Q[t] may be written uniquely (up to multiplication by 1) as a product of its content
( Q ) times a primitive polyno
Introduction to: Bilinear Forms, Linear groups, and Group
Denition of a bilinear mapping U V W . Discuss dot-product. Rn Rn R. Examples
over any eld;
V V F
F n F n F.
This is determined by a matrix A.
(x, y) x, y y = xt
Finite generation of modules; noetherian rings
Prop 5.13 of [A]: The following conditions on an R-module M are equivalent:
1. Every sub-R-module of M is nitely generated.
2. Ascending Chain Condition: There is no innite strictly ascend
Intro to the Fundamental Theorem of Galois Theory
Reading: [Artin] pp. 493-508, 537-547
Homomorphisms over k map elements that are algebraic over k
to their k-conjugates; intro to Galois groups
Let k be a eld. in these notes it will be our groundeld.
Do this in the positive denite real symmetric and Hermitian situations. There is only one positive denite real symmetric (bilinear) form in every dimension; also only one positive denite
Hermitian form. Complete the on-g