MTWRF 11:30-12:25 UH (University Hall) 074
Instructor: Sasha Leibman
office:
MW (Math Tower) 406
e-mail:
leibman.1@osu.edu
phone: 614-620-7767
Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition
Link to Modules
Lecture notes: Galois theory (version of 3/25)
Homework:
| Homework 8 | – due by Tuesday, March 24. Solutions |
| Homework 9 | – due by Tuesday, March 31. |
Calendar: [LN=Lecture Notes, TB=Text Book]
| March 6: |
Introduction to the Galois theory
Fields, prime subfields, characteristic [LN 1.1, TB 13.1] |
| March 9: |
Extensions and subextensions of fields.
Towers and composites of subextensions
[LN 1.2, TB 13.1]
Finite extensions, towers of finite extension [LN 1.3, TB 13.1] Simple extensions, algebraic and transcendental elements, minimal polynomials [LN 1.4, TB 13.2] |
| March 10: |
Methods of computing the minimal polynomial
[LN 1.4, TB 13.2]
Towers of simple extensions [LN 1.5, TB 13.2] The composite of two finite extensions [LN 1.6, TB 13.2] |
| March 11: | Quadratic and biquadratic extensions [LN 1.7, TB exercises 13.2.7-9] |
| March 12: |
Algebraic extensions
[LN 1.8, TB 13.2]
Adjoining roots of polynomials [LN 2.1, TB 13.1] |
| March 13: | Splitting fields [LN 2.2, TB 13.4] |
| March 23: |
Uniqueness of splitting fields
[LN 2.2, TB 13.4]
Algebraic closure [LN 2.3, TB 13.4] |
| March 24: | Exercises from TB Sections 13.2,4 |
| March 25: | Separable and inseparable polynomials and extensions. [LN 2.4, TB 13.5] |
| March 26: | Roots of unity, cyclotomic extensions and cyclotomic polynomials (LN 3.1.1-10, TB 13.6) |
| March 27: |
Cyclotomic polynomials are irreducible
(LN 3.1.11-13, TB 13.6)
Finite fields (LN 3.2.1-2, TB 13.5, 14.3) |
Plans:
Subfields of finite fields and irreducible polynomials in Fp[x]
(LN 3.2.3-7, TB 13.5, 14.3)
Embeddings of finite extensions
(LN 4.1, TB 14.1-2)