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
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
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 gi
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 abe
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
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
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. Ascend
Intro to the Fundamental Theorem of Galois Theory
Reading: [Artin] pp. 493-508, 537-547
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Homomorphisms over k map elements that are algebraic over k
to their k-conjugates; intro to Galois groups
Let
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 onl
Intro to the Fundamental Theorem of Galois Theory
1
Homework Set due April 1
1. Artin Page 575 Exercises: 1, 6, 10, 12, 18.
2. Artin Page 576 (Primitive Elements ) Exercises: 2, 3,6,
3. Artin Page 577