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 4/27)
Final exam is scheduled for Tuesday, May 5, from 10am to 12pm, at UH 074. It will cover only Galois theory and consist of 6–7 problems. The use of textbooks, lecture notes, and personal notes is permitted, the use of electronic devices is prohibited.
| Homework 8 | – due by Tuesday, March 24. Solutions |
| Homework 9 | – due by Tuesday, March 31. Solutions |
| Homework 10 | – due by Tuesday, April 7. Solutions |
| Homework 11 | – due by Tuesday, April 14. Solutions |
| Homework 12 | – due by Tuesday, April 21. Solutions |
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) |
| March 30: |
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) |
| March 31: | Normal extensions (LN 4.2, TB 13.4) |
| April 1: |
Galois extensions and Galois groups
(LN 4.3, TB 14.1-2)
Examples of Galois groups (LN 4.5.1-3, TB 14.1-2) |
| April 2: | Examples of Galois groups (LN 4.5.3-8, TB 14.1-2) |
| April 3: | Composites and towers of separable extensions (LN 4.4) |
| April 6: | The fundamental theorem of the Galois theory (LN 4.6.1-5, TB 14.2) |
| April 7: |
The fundamental theorem of the Galois theory – full version
(LN 4.6.6, TB 14.2)
Examples of diagrams of subextensions and the corresponding Galois groups (LN 4.7, exercises from TB 14.2) |
| April 8: | The Galois groups of composites (LN 5.1-3, TB 14.4) |
| April 9: | The Galois groups of composites (LN 5.3, TB 14.4) |
| April 10: |
Free composites of Galois extensions
(LN 5.4, TB 14.4)
Abelian extensions (LN 6.3) The Galois group of a tower of Galois extensions (LN 5.5, TB 14.4) |
| April 13: |
More methods of finding the minimal polynomial
(LN 6.1)
The norm of algebraic elements (LN 6.2, TB exercise 14.2.17) |
| April 14: |
The norm of algebraic elements
(LN 6.2, TB exercise 14.2.17)
Subextensions of real radical extensions (LN 6.4.1, TB exercises 14.7.4-5) |
| April 15: |
The Galois group of xn-a with a>0
(LN 6.4.2-3, TB exercise 14.7.6)
The theorem on a primitive element (LN 6.5, TB 14.4) |
| April 16: |
p-extensions
(LN 6.6)
The fundamental theorem of algebra (LN 6.7, TB 14.4) |
| April 17: | Constructions with ruler and compass (LN 6.8.5-9, TB 13.3, 14.5) |
| April 20: |
Linear independence of square roots of square-free integers
(LN 6.9)
Symmetric polynomials and rational functions (LN 6.10, 14.6) |
| April 21: | Radical, polyradical, cyclic, and polycyclic extensions (LN 7.1-3, 14.7) |
| April 22: |
Solvability of polynomials in radicals
(LN 7.4, 14.7)
The alternating group and the discriminant (LN 7.5, 14.7) |
| April 23: |
Galois group and solution in radicals of cubics
(LN 7.6, 14.6-7)
Casus irreducibilis (LN 7.6.6, 14.6) |
| April 24: | Galois group and solution in radicals of quartics (LN 7.7, 14.6-7) |
| April 27: | Examples and methods of computation of Galois groups, Dedekind's theorem (LN 7.8) |
Plans:
Final exam on Tuesday, May 5, from 10am to 12pm, at UH 074.