Math 200 B: Algebra II

Winter 2013

Lectures: M-W-F 12:00 PM--12:50 PM  APM 5402
Office Hour: Th 1:00 PM--3:00 PM APM 7230
TA information: Michael Kasa mkasaucsd edu
TA Office hour: Tue 3:00 PM--4:00 PM  APM 6452


  • 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.


We might have minor deviations.

  • 01/07/13 Section 10.1-10.2: 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 HomR(M,N) and EndR(M) and reinterpreting the definition of an R-mod, 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 direct-summand.
  • 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; p-primary components;
  • 01/28/13 Section 12.1-12.2: 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; Cayley-Hamilton 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 (R-Mod --> S-Mod) 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 (R-Mod --> (Abelian) R-Mod); 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 A-Mod --> Abelian is; HomA(M,_) functor; HomA(M,_) is left exact; Projective modules; M is projective iff HomA(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: HomA(_,M) (contravariant) functor; HomA(_,M) is right exact; Injective modules; M is injective iff HomA(_,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 Fp and its group of automorphisms;
  • 03/11/13 Section 14.4-14.5: 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.


  • 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, {Mi}i ∈ I be a family of R-mod and N be also R-mod. Prove that
      • HomR(⊕i ∈ I Mi, N)≅ ∏i ∈ I HomR(Mi, N),
      • HomR(N,∏i ∈ I Mi)≅ ∏i ∈ I HomR(N,Mi),
      • With an example show that HomR(∏i ∈ I Mi, N) is not isomorphic to ∏i ∈ I HomR(Mi, N) or ⊕i ∈ IHomR(Mi, 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:

  • Here is the midterm. (R in the last part of the first problem should be changed to Rop.)
  • Here is the final.