MTWRF 11:30-12:25 JR (Journalism Building) 353
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 the first part "Modules"
Lecture notes:
Galois theory
The final exam is going to be on Friday, April 26, 12-2pm. (If this day/time is inconvenient to you let me know.) The exam will only cover the second part of the course, the Galois theory; it will be open book(s)/notes, but internet won't be allowed. Here are some (rather random) practice problems with solutions.
Homework:
| Homework 8 | – due by Tuesday, March 26. Solutions |
| Homework 9 | – due by Tuesday, April 2. Solutions |
| Homework 10 | – due by Tuesday, April 9. Solutions |
| Homework 11 | – due by Tuesday, April 16. Solutions |
Calendar: [LN=Lecture Notes, TB=Text Book]
| March 6: | Introduction to Galois theory |
| March 7: |
Fields, prime subfields, characteristic
(LN 1.1, TB 13.1)
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) |
| March 8: | Simple extensions, algebraic and transcendental elements, minimal polynomials (LN 1.4, TB 13.2) |
| March 18: |
Towers of simple extensions
(LN 1.5, TB 13.2)
The composite of two finite extensions (LN 1.6, TB 13.2) |
| March 19: | Quadratic and biquadratic extensions (LN 1.7, TB exercises 13.2.7-9) |
| March 20: |
Algebraic extensions
(LN 1.8, TB 13.2)
Adjoining roots of polynomials (LN 2.1, TB 13.1) |
| March 21: | Splitting fields (LN 2.2, TB 13.4) |
| March 22: | Algebraically closed fields and the algebraic closure of a field (LN 2.3, TB 13.4) |
| March 25: |
Exercises from TB Sections 13.2,4
Separable and inseparable polynomials and extensions. (LN 2.4.1-7, TB 13.5) |
| March 26: |
Perfect fields and the Frobenius homomorphism
(LN 2.4.8-14, TB 13.5)
Cyclotomic extensions and cyclotomic polynomials (LN 3.1.5-13, TB 13.6) |
| March 27: | Finite fields (LN 3.2, TB 13.5, 14.3) |
| March 28: |
Embeddings of finite extensions
(LN 4.1, TB 14.1-2)
Normal extensions (LN 4.2, TB 13.4) |
| March 29: | Galois extensions and Galois groups (LN 4.3, TB 14.1-2) |
| April 1: | Examples of Galois groups (LN 4.5.1-6, TB 14.1-2) |
| April 2: | Examples of Galois groups (LN 4.5.6-8, TB 14.1-2) |
| April 3: |
Composites and towers of separable extensions
(LN 4.4)
The fundamental theorem of the Galois theory – short version (LN 4.6.1-5, TB 14.2) |
| April 4: | The fundamental theorem of the Galois theory – full version (LN 4.6.6, 4.7.1-2, TB 14.2) |
| April 5: | Exercises from TB Sections 14.1,2 |
| April 8: |
The Galois groups of composites
(LN 5.1-3, TB 14.4)
Free composites of Galois extensions (LN 5.4, TB 14.4) |
| April 9: |
The Galois groups of towers of Galois extensions
(LN 5.5, TB 14.4)
Linear independence of square roots of square-free integers (LN 6.9) |
| April 10: |
More methods of finding the minimal polynomial
(LN 6.1)
The norm of algebraic elements (LN 6.2, TB exercise 14.2.17) Abelian extensions (LN 6.3) Subextensions of real radical extensions (LN 6.4.1, TB exercises 14.7.4-5) |
| April 11: |
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) p-extensions (LN 6.6) |
| April 12: |
The fundamental theorem of algebra
(LN 6.7, TB 14.4)
Constructions with ruler and compass (LN 6.8.1-4, TB 13.3, 14.5) |
| April 15: | Constructions with ruler and compass (LN 6.8.5-9, TB 13.3, 14.5) |
| April 16: |
Symmetric polynomials and rational functions
(LN 6.10, 14.6)
Radical, polyradical, cyclic, and polycyclic extensions (LN 7.1-3, 14.7) |
| April 17: |
Solvability of polynomials in radicals
(LN 7.4, 14.7)
The alternating group and the discriminant (LN 7.5, 14.7) |
| April 18: | Galois group and solution in radicals of cubics (LN 7.6, 14.6-7) |
| April 19: |
Casus irreducibilis
(LN 7.6.6, 14.6)
Galois group and solution in radicals of quartics (LN 7.7, 14.6-7) |
| April 22: |
Dedekind's theorem
(LN 7.8.2-5)
Practice problems |
Plans:
Final exam on Friday, April 26, 12-2pm.