Fields and their automorphisms
Reading: [Artin] pp. 493-508.
1
Basic denitions
Concepts of algebraic, transcendental for elements, eld extensions; discuss. Finite degree implies
algebraic. Examples.
2
Fundamental construction
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.
1
Recall
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
Characters, continued
1
Orthogonality Theorem
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
1
One-dimensional representations
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
1
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
1
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
representations
1
Bilinear forms
Denition of a bilinear mapping U V W . Discuss dot-product. Rn Rn R. Examples
over any eld;
V V F
or
F n F n F.
This is determined by a matrix A.
(x, y) x, y y = xt
Finite generation of modules; noetherian rings
1
Finite generation
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
1
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.
De
Characters
1
Orthogonal Complement
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