everything.study
18.702

Algebra II

3-0-9 units · Undergraduate

Continuation of 18.701 . Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Spring 2011

Topics

  • Group representations
  • Unitary representations
  • Characters
  • One-dimensional characters
  • The regular representation
  • Characters (cont.)
  • Rings, ring homomorphisms
  • Ideals, quotient rings, correspondence theorem
  • Maximal ideals, prime ideals, fractions
  • Gauss’ Lemma
  • Unique factorization
  • Factoring integer polynomials
  • Gauss primes
  • Quadratic integers
  • Ideal factorization
  • Prime ideals
  • Ideal classes
  • Integer matrices
  • Free modules
  • Presenting modules
  • Hilbert Basis Theorem
  • Structure of abelian groups
  • Linear Operators
  • Algebraic elements, degree
  • Ruler and compass
  • Adjoining elements
  • Finite fields
  • Primitive elements
  • Symmetric functions, discriminant
  • Splitting fields, the Galois group
  • Fixed fields, Galois extensions
  • The main theorem
  • Cubic and quartic equations
  • Quartic equations (cont.)
  • Roots of unity, Kummer extensions
  • Quintic equations

Readings

  • Artin, M. Algebra (2nd Edition) . Addison Wesley, 2010. ISBN: 9780132413770.
    Listed reading
  • SES #: 1 · TOPICS: Group representations · READINGS: Chapter 10, sections 1, and 4 · EXERCISES: Chapter 10, 1.1, and 1.2
    Reading assignment
  • SES #: 2 · TOPICS: Unitary representations · READINGS: Chapter 10, sections 2, and 3 · EXERCISES: Chapter 10, 2.1, 2.2, and 3.4
    Reading assignment
  • SES #: 3 · TOPICS: Characters · READINGS: Chapter 10, section 4 · EXERCISES: Chapter 10, 4.1, 4.3a, b, and 4.8
    Reading assignment
  • SES #: 4 · TOPICS: One-dimensional characters · READINGS: Chapter 10, section 5 · EXERCISES: Chapter 10, 5.1, and 5.3
    Reading assignment
  • SES #: 5 · TOPICS: The regular representation · READINGS: Chapter 10, section 6 · EXERCISES: Chapter 10, 6.1, and 6.10
    Reading assignment
  • SES #: 6 · TOPICS: Characters (cont.) · READINGS: Chapter 10, sections 7, and 8 · EXERCISES: Chapter 10, 7.1, 7.2, and 7.6
    Reading assignment
  • SES #: 7 · TOPICS: Rings, ring homomorphisms · READINGS: Chapter 11, sections 1, 2, and 3 · EXERCISES: Chapter 11, 1.1, 1.5, 1.8, and 1.9
    Reading assignment
  • SES #: 8 · TOPICS: Ideals, quotient rings, correspondence theorem · READINGS: Chapter 11, sections 4, and 5 · EXERCISES: Chapter 11, 3.12, 3.13, 4.1, and 4.2
    Reading assignment
  • SES #: 9 · TOPICS: Maximal ideals, prime ideals, fractions · READINGS: Chapter 11, sections 8, and 9 · EXERCISES: Chapter 11, 6.1, 7.1, and 8.3
    Reading assignment
  • SES #: 10 · TOPICS: Gauss’ Lemma · READINGS: Chapter 12, section 3 · EXERCISES: Chapter 12, 2.3, 2.7, and 3.2
    Reading assignment
  • SES #: 11 · TOPICS: Unique factorization · READINGS: Chapter 12, sections 1, and 2 · EXERCISES: Chapter 12, 1.1, 1.5, 2.1, and 2.2
    Reading assignment
  • SES #: 12 · TOPICS: Factoring integer polynomials · READINGS: Chapter 12, section 4 · EXERCISES: Chapter 12, 4.1a, 4.6, 4.7, and 4.11
    Reading assignment
  • SES #: 13 · TOPICS: Gauss primes · READINGS: Chapter 12, section 5 · EXERCISES: Chapter 12, 5.1, 5.2a, and 5.3
    Reading assignment
  • SES #: 14 · TOPICS: Quadratic integers · READINGS: Chapter 13, section 1 · EXERCISES: Chapter 13, 1.1, 1.2, and 1.3
    Reading assignment
  • SES #: 15 · TOPICS: Ideal factorization · READINGS: Chapter 13, sections 2, and 3 · EXERCISES: Chapter 13, 2.1, 3.1, 3.2, and 3.3
    Reading assignment
  • SES #: 16 · TOPICS: Prime ideals · READINGS: Chapter 13, sections 5, and 6 · EXERCISES: Chapter 13, 5.3, 6.1, and 6.2
    Reading assignment
  • SES #: 17 · TOPICS: Ideal classes · READINGS: Chapter 11, sections 9, and 10 · EXERCISES: Chapter 13, 7.1, 7.2, and 8.2
    Reading assignment
  • SES #: 18 · TOPICS: Integer matrices · READINGS: Chapter 14, sections 1, and 2 · EXERCISES: Chapter 14, 1.1, 2.1, and 2.4
    Reading assignment
  • SES #: 19 · TOPICS: Free modules · READINGS: Chapter 14, sections 3, and 4 · EXERCISES: Chapter 14, 3.2, 4.1a, and 4.3
    Reading assignment
  • SES #: 20 · TOPICS: Presenting modules · READINGS: Chapter 12, section 5 · EXERCISES: Chapter 14, 5.1, and 5.2
    Reading assignment
  • SES #: 21 · TOPICS: Hilbert basis theorem · READINGS: Chapter 14, section 6 · EXERCISES: Chapter 11, 6.1, 6.2, and M.1
    Reading assignment
  • SES #: 22 · TOPICS: Structure of abelian groups · READINGS: Chapter 14, section 7 · EXERCISES: Chapter 14, 7.1, 7.2, and 7.5
    Reading assignment
  • SES #: 23 · TOPICS: Linear operators · READINGS: Chapter 14, section 8 · EXERCISES: Chapter 14, 8.1, 8.3, and 8.4
    Reading assignment
  • SES #: 24 · TOPICS: Algebraic elements, degree · READINGS: Chapter 15, sections 1, and 2 · EXERCISES: Chapter 15, 1.1, 1.3, and 2.1
    Reading assignment
  • SES #: 25 · TOPICS: Ruler and compass · READINGS: Chapter 13, section 5 · EXERCISES: Chapter 15, 5.1, 5.2, and 5.4
    Reading assignment
  • SES #: 26 · TOPICS: Adjoining elements · READINGS: Chapter 11, section 5, chapter 15, section 6 · EXERCISES: Chapter 11, 5.2, and 5.3, chapter 15, 6.1, and 6.3
    Reading assignment
  • SES #: 27 · TOPICS: Finite fields · READINGS: Chapter 15, section 7 · EXERCISES: Chapter 15, 7.1, 7.2, and 7.13
    Reading assignment
  • SES #: 28 · TOPICS: Primitive elements · READINGS: Chapter 15, section 8 · EXERCISES: Chapter 15, 8.1, and 8.2
    Reading assignment
  • SES #: 29 · TOPICS: Symmetric functions, discriminant · READINGS: Chapter 16, sections 1, and 2 · EXERCISES: Chapter 16, 1.1a, b, e, 2.1, and 2.2
    Reading assignment
  • SES #: 30 · TOPICS: Splitting fields, the Galois group · READINGS: Chapter 16, sections 3, and 4 · EXERCISES: Chapter 16, 3.2, and 4.1
    Reading assignment
  • SES #: 31 · TOPICS: Fixed fields, Galois extensions · READINGS: Chapter 16, sections 5, and 6 · EXERCISES: Chapter 16, 5.1b, c, 6.1, and 6.2
    Reading assignment
  • SES #: 32 · TOPICS: The main theorem · READINGS: Chapter 16, section 7 · EXERCISES: Chapter 16, 7.1, 7.3, 7.6, and 7.7
    Reading assignment
  • SES #: 33 · TOPICS: Cubic and quartic equations · READINGS: Chapter 16, sections 8, and 9 · EXERCISES: Chapter 16, 8.2a, b, c, 9.1, 9.3, and 9.6
    Reading assignment
  • Chapter 16, section 9
    Listed reading
  • Chapter 16, 9.12a, b, and 9.16
    Listed reading
  • SES #: 35 · TOPICS: Roots of unity, Kummer extensions · READINGS: Chapter 16, sections 10, and 11 · EXERCISES: Chapter 16, 10.1, 10.3, and 11.1
    Reading assignment
  • SES #: 36 · TOPICS: Quintic equations · READINGS: Chapter 16, section 12 · EXERCISES: Chapter 16, 12.1, 12.2, and 12.7
    Reading assignment

Prerequisites & course information

Prereq: 18.701 U (Spring) 3-0-9 units

Sources