Groups, Rings, and Vector Spaces II: Topics list

Topics & Pacing

The following is a rough list of arguments covered in class.

Suggested homework problems have been moved to the Homework page.

Apr. 2

Submodules of free modules. Classification of modules over a PID (statement).

Mar. 30

Koszul complexes (from the problems). Submodules of free modules over a PID.

Mar. 28

Resolutions. Presentations and cokernels via matrices.

Mar. 26

Torsion and torsion-free modules. Finite generation and presentation.

Mar. 23

Euler characteristic and the Grothendieck group.

Mar. 21

Proof that A with entries in a commutative ring is invertible if and only if det A is a unit. Cramer's rule. Row and column spaces, rank and nullity.

Mar. 19

Determinants. Proof that A with entries in a field is invertible if and only if det A is nonzero. Cofactors and adjoint matrix.

Mar. 16

• Gaussian elimination over Euclidean Domains. Smith normal form.
• Linear systems.

Mar. 14

Equivalence of matrices. Elementary operations and Gaussian elimination over fields.

Mar. 12

Homomorphisms of free modules. Matrices. Change of basis.

Mar. 2

Vector spaces. Rank and dimension.

Feb. 29

Free modules. Linear independence and bases.
Homework: VI.1: #1, 2, 4, 6.

Feb. 27

Chinese Remainder Theorem. Gaussian integers.
Homework: V.6: #1, 5, 6, 7, 9, 13, 14, 17.

Feb. 24

Irreducibility in Q, R, and C. Eisenstein's criterion.
Homework: V.5: #4, 6, 11, 13, 20, 21, 22, 23, 24.

Feb. 22

Reducibility and roots. Field extensions and adding roots. Algebraically closed fields.

Feb. 20

Localization. Local rings
Homework: Problems in V.4.

Feb. 17

Midterm 1

Feb. 15

Localization

Feb. 13

Problems.

Feb. 10

Fields of fractions. R UFD => R[x] UFD.
Homework: V.4: #6, 7-12 (also in class), 15, 16, 17, 22.

Feb. 8

Primitivity and content. Fields of fractions.

Feb. 6

• Posets. Zorn's Lemma. Homework: V.3: #13, 14.
• Valuations. Euclidean domain => PID. Euclidean Algorithm.
  Homework: V.2: #12, 14--21.

Feb. 3

Characterizations of UFDs. PID => UFD.
Homework: V.2: #4, 6, 7, 9, 11.

Feb. 1

Divisors and gcd in integral domains. Multisets of irreducible factors.

Jan. 30

Primes and irreducibles. Factorization, UFDs.
Homework: V.1: #14, 17.

Jan. 27

Problems.

Jan. 25

Noetherian rings. Hilbert's basis theorem.
Homework: V.1: #2, 5, 7, 8.

Jan. 23

Classification theorem of finite abelian group. Elementary divisors. Invariant factors.
Homework: IV.6: #1, 4, 5, 7, 8, 12, 14.

Jan. 20

Semidirect products.
Homework: IV.5: #1 (see IV.3.10), 7, 10, 12, 13, 14, 15.

Jan. 18

• Short exact sequences of groups. Splittings.
• Simplicity of A_n, finished.

Jan. 16

No class: MLK Day

Jan. 13

Conjugacy classes in S_n and A_n. Simplicity of A_n, non-solvability of S_n.
Homework: IV.4: #6, 7, 9, 12, 17.

Jan. 11

Conjugacy classes and types. Even and odd permutations. Transpositions. Alternating groups.

Jan. 9

• Symmetric groups. Cycles. Partitions. Types. Young diagrams.
  Homework: IV.4: #3, 4.
• Solvability (finished)

Jan. 6

Composition factors, Schreier Theorem. Solvability.
Homework: IV.3: #2, 4, 13, 15.

Jan. 4

Normal and composition series. Jordan-Hoelder Theorem.