
Winter 2013
Lectures:

MWF

12:00 PM12:50 PM

APM 5402

Office Hour:

Th

1:00 PM3:00 PM

APM 7230

TA information:


Michael Kasa

mkasaucsd edu

TA Office hour:

Tue

3:00 PM4:00 PM

APM 6452


Book
 D. S. Dummit and R. M. Foote, Abstract algebra. (The main textbook)
 D. Hungerford, Algebra.
 P. Morandi, Field and Galois theory.
 I. M. Isaacs, Algebra.

Schedule
We might have minor deviations.
 01/07/13 Section 10.110.2: Definition of modules, examples,
direct product and direct sum of modules, definition of submodule and quotient modules,
the first isomorphism theorem.
 01/09/13 Section 10.2: various structures on Hom_{R}(M,N) and End_{R}(M) and reinterpreting the definition of an Rmod, F[x]modules.
 01/11/13 Section 10.3: universal properties of direct sum and direct product of modules, simple and semisimple modules.
 01/14/13 Not from the text: semisimple modules and directsummand.
 01/16/13 Not from the text: semisimple modules (continue) and Noetherian modules.
 01/18/13 Section 12.1: modules over PIDs
 01/21/13 No class.
 01/23/13 Section 12.1: Submodules of a free module over a PID; rank of a module over a domain;
 01/25/13 Section 12.1: Structure of f.g. modules over a PID (existence); Invariant factors; Elementary divisors; pprimary components;
 01/28/13 Section 12.112.2: Structure of f.g. modules over a PID (uniqueness); Eigenvalues, characteristic polynomial and minimal polynomials of a linear transformation; Looking at V as an F[x]module via T; Connection between Ann(V) and the minimal polynomial of T;
 01/30/13 Section 12.2: Invariant factors of a linear transformation and their connection with the characteristic polynomial and the minimal polynomial; CayleyHamilton theorem, more on similarities of matrices;
 02/01/13 Section 12.3: Jordan canonical form; Nilpotent matrices; tensor product;
 02/04/13 Section 10.4: Base change (RMod > SMod) and its universal property; rank_R(M)=dim_F(M tensor F) where F is the quotient field of the domain R; Base change and direct sum;
 02/06/13 Section 10.4: Tensor product of two modules (RMod > (Abelian) RMod); Universal property; Z/nZ tensor Z/mZ (over Z); R/I tensor N over R;
 02/08/13 Section 10.5: Exact sequences; Short exact sequences; Split short exact sequences; Equivalency of short exact sequences;
 02/11/13 Midterm
 02/13/13 Section 10.5: What a (covariant functor) from AMod > Abelian is; Hom_{A}(M,_) functor; Hom_{A}(M,_) is left exact; Projective modules; M is projective iff Hom_{A}(M,_) is exact iff it has a certain lifting property iff any short exact sequence ending in M splits iff M is a direct summand of a free module.
 02/15/13 Section 10.5: Hom_{A}(_,M) (contravariant) functor; Hom_{A}(_,M) is right exact; Injective modules; M is injective iff Hom_{A}(_,M) is exact iff M has a certain lifting property iff any short exact sequence starting with M splits iff M is a direct summand of N when M is a submodule of N.
 02/18/13 No class.
 02/19/13 Section 10.5: Baer's crteria for injective modules; Connection between the tensor product and Hom; Tensor product is right exact; flat modules; M is flat iff M tensor _ over A is exact iff f tensor id._{M} is injective when f is injective iff injection induces an embedding of a tensor M into A tensor M for any ideal a of A. (Extra meeting)
 02/20/13 Section 13.1: Field extensions; Solving a polynomial equation in an extension; Dimension of F[a]=F(a) over F; F(a) is isomorphic to F'(a') if F'=t(F), P(a)=0 and t(P)(a')=0 where is irreducible over F;
 02/22/13 Section 13.2, 13.4: Finite extensions are algebraic; [L:F]=[L:K][K:F]; The set of algebraic elements is a subfield; Splitting field; Uniqueness of splitting fields;
 02/25/13 Section 13.4, 14.1: Algebraic closure; algebraically closed; Aut(F); Aut(E/F) where E is the splitting field of a polynomial f; Galois extensions.
 02/27/13 Section 14.2: Characters are linearly independent; E/Fix(G) is Galois if G is a subgroup of Aut(E); Aut(E/F) is at most [E:F];
 03/01/13 Section 14.2, 13.5: Separable and normal extensions; E/F is Galois iff E/F is separable and normal; Fundamental Theorem of Galois theory;
 03/04/13 Section 14.2: Fundamental Theorem of Galois theory (continue);
 03/06/13 Section 14.3, Not from the text: Finite fields;
 03/08/13 Algebraic closure of F_{p} and its group of automorphisms;
 03/11/13 Section 14.414.5: Infinite Galois theory; Primitive element theorem;
 03/13/13 Section 14.7: Solvable and radical extensions.
 03/15/13 Section 14.7: Insolvability of the quintic.

Assignments.
 Due 01/13:
 Section 10.1: 8, 21.
 Section 10.2: 6, 12.
 Section 10.3: 2, 11.
 Let R be a ring with 1, {M_{i}}_{i ∈ I} be a family of Rmod and N be also Rmod.
Prove that
 Hom_{R}(⊕_{i ∈ I} M_{i}, N)≅ ∏_{i ∈ I} Hom_{R}(M_{i}, N),
 Hom_{R}(N,∏_{i ∈ I} M_{i})≅ ∏_{i ∈ I} Hom_{R}(N,M_{i}),
 With an example show that Hom_{R}(∏_{i ∈ I} M_{i}, N) is not isomorphic to
∏_{i ∈ I} Hom_{R}(M_{i}, N) or
⊕_{i ∈ I}Hom_{R}(M_{i}, N).
 Due 01/20:
 No problem for this week!
 Due 01/27:
 Due 02/03:
 Due 02/17:
 Due 02/24:
 Due 03/03:
 Due 03/10:
 Due 03/17:

Exams.
 Here is the midterm. (R in the last part of the first problem should be changed to R^{op}.)
 Here is the final.
