Summary of topics covered.

  • 1. Basics on commutative rings, ideals, modules. Brief treatment of field of fractions of integral domain and of inverting a single element in an arbitrary commutative ring.
  • 2. Factorization in PID's. Euclidean domains.
  • 3. Content, Gauss Lemma, Eisenstein irreduciblity, UFD's.
  • 4. Finitely generated modules over a PID.
  • 5. Jordan canonical form.
  • 6. Cayley Hamilton Theorem for finitely generated free modules over a commutative ring. (Brief treatment.)
  • 7. Noetherian rings, submodules of finitely generated modules over a Noetherian ring, proof of Hilbert basis theorem.
  • 8. Basics on field extensions.
  • 9. Basics on finite fields.
  • 10. Basic Galois theory, including proof of fundamental theorem and discussion of Galois groups of low degree polynomials.
  • 11. Bare bones Kummer theory, cyclic extensions in the presence of roots of one.
  • 12. Sylow theorems. Solvable finite groups. Simplicity of the alternating group, A_5.
  • 13. Very basic treatment of cyclotomic fields, not including quadratic subfields.
  • 14. Solvability by radicals if and only if solvable Galois group (under appropriate hypotheses on the characteristic).
  • 15. Finite separable extensions, finite purely inseparable extensions, finite normal extensions.
  • 16. Transcendence degree and transcendence basis.
  • 17. Lueroth's theorem.
  • 18. One lecture on derivations and finitely generated field extensions in characteristic zero.
  • 19. Categories and functors: The concepts were defined and the language was used a little (eg. category of k-algebras, the torsion submodule of a module over an integral domain as an example of a functor). The notions of cokernel and exact sequence were perhaps mentioned once each in the course.

    Topics which might conceivably be treated in Math 251, but which were not covered in fall 2002.

  • 1. Tensor products, multilinear maps.
  • 2. Integral extensions, integral closure, Hilbert's Nullstellensatz.
  • 3. Galois theory for infinite algebraic extensions.
  • 4. Multiplicative systems in a ring and a treatment of localization which is adequate for a commutative algebra course. (See 1 under topics covered.)
  • 5. Linearly disjoint field extensions. Separating transcendence bases.
  • 6. Discrete valuations, absolute values.
  • 7. Hilbert's Theorem 90, group cohomology.
  • 8. Semi-direct products, nilpotent groups, structure of finite groups.
  • 9. Additive and abelian categories. The Hom functor.

    Math 251 Algebra Fall 2002: Homework.

    1. Homework for Monday September 2:

  • Read Artin: Sections 10.1- 10.4 (Go lightly on formal definition of integers.)
  • Exercises:
  • 10.1 1,2,3,4,5, 8a,c, 10, 11, 12, 14
  • 10.2 6,7
  • 10.3 1,2, 8, 17, 25a, 34
  • 10.4 7,8
  • Instructor will consider all feedback on amount and difficulty of homework.

    2. Homework for Monday September 9:

  • Read in Artin: 10.6, 11.1, 11.2
  • Exercises:
  • 10.5 2,6,8,16
  • 10.6 2,3,6
  • 10.7 2(a) and (b), 11
  • 11.1 4(a), 5,6,8,11,12
  • 11.2 5

    3. Homework for Monday September 16:

  • Read in Artin: 11.3-11.5
  • Handout on the binary quadratic form x^2-2y^2.

    4. Homework for Monday September 23:

  • Read in Artin: 12.1-12.6
  • 11.2 7,8,10
  • 11.4 2, 4, 8, 9 d,e,f
  • 11.5 1,3
  • 12.1 1, (think about 3 and 12, but don't hand in the solutions)
  • 12.2 4,5
  • 12.5 1
  • 12.6 1, 2, 3, 9

    5. Homework for Monday September 30:

  • Please read the handout on Hermite normal form. Please read Artin 12.7
  • Please do the exercises on Hermite normal form.
  • Please do the late September exercises.
  • Please work the following problems in Artin:
  • 12.6 4,5
  • 12.7 1, 2, 3, 4, 5, 16, 19

    6. Homework for Monday October 7 :

  • Please review the material on Noetherian rings in Artin 12.5.
  • Please read Artin 13.1-13.3
  • Please work the exercises on the handout
  • Please work in Artin exercises
  • 12.5 5
  • 13.1 3
  • 13.2 3
  • 13.3 1, 2, 3, 4, 7, 8, 14.

    7. Homework for Monday October 21:

  • Please read Artin 13.5-13.6.
  • Please work problems 1-17 on the handout on factoring polynomials with coefficients in finite fields.
  • Please review with care the updated handout on the computer software GP.
  • Please work problems 18-22 on the handout.
  • Please read Artin 14.1.
  • Please work problems 23-27 on the handout.

    8. Homework for Monday October 28:

  • Please read Artin 14.1-14.4.
  • Please work the following exercises in Artin:
  • 14.1 6,8,10,12,13,16,18
  • 14.2 9
  • 14.3 4

    9. Homework for Monday November 4:

  • Please read Artin 14.5-14.6.
  • Please work the following exercises in Artin:
  • 14.5 1, 2, 3, 4, 8, 9, 10, 11
  • 14.6 5,6

    10. Homework for Monday November 11:

  • Please read Artin 14.7-14.8
  • Please work the following exercises in Artin:
  • 14.8 4,6,9,10.
  • 14. Miscellaneous problems: 11
  • Handout problems on cyclotomic polynomials
  • Three handout problems (you choose which) on finite degree field extensions.
  • Problem 14.5 9(c) in Artin revisited--see backside of handout.

    11. Homework for Monday November 18:

  • Please read Artin 14.9
  • Please work exercises 14.9 1, 2, 6, 8
  • Five problems on finite degree field extensions different from the ones you did last time.

    12. Homework for Monday November 25:

  • Review Artin 5.5, 5.6, 5.7 as needed. Please read Artin 6.4.
  • Please work in Artin section 6.4 problems 1,2,4,15.
  • Handout on group theory.

    13. Homework for Monday December 2:

  • Please read section 13.8 in Artin.
  • Please work 10 problems of your choice from the following:
  • Artin 13.8 problem 3.
  • Handout on non-parametrizability.
  • Handout on derivations.

    14. Happy new year handout:

  • Handout on partial fraction decomposition