Week

Date

Topics

Reading

Homework

1

Mon 06 Jan

History of solving polynomial equations. The complex numbers.



Wed 08 Jan

Review of ring theory: Euclidean domains, PID, UFD. Irreducible
polynomials and ideals in polynomial rings.

DF 9.2
FT 711

Fri 10 Jan

Roots. Fundamental
Theorem of Arithmetic. Reduction
mod p.
Irreducibility criteria for polynomials. Irreducible polynomials over
finite fields. Eisenstein's criterion.

DF 9.29.5
FT 1113

2

Mon 13 Jan

Gauss's Lemma and
primitive polynomials.
Tower law for degrees.

DF 9.3, 13.1
FT 1317

Problem Set #1

Wed 15 Jan

Finitely generated extensions.
Simple extensions. Classification of simple extensions.
Transcendental and algebraic elements. Minimal polynomial.

DF 13.2
FT 1519

Fri 17 Jan

Algebraic extensions, continued. Compass and straightedge. Constructible numbers. Pythagorean closure.

DF 13.3
FT 2123

3

Mon 20 Jan

No class: Martin Luther King, Jr. Day

Problem Set #2

Wed 22 Jan

Construction impossibility proofs, continued. Regular ngons. Splitting fields.

DF 13.34
FT 2830

Fri 24 Jan

Splitting fields.

DF 13.4
FT 2425

4

Mon 27 Jan

Separability.

DF 13.5
FT 3033

Problem Set #3

Wed 29 Jan

Multiple roots. Embeddings.

DF 13.5
FT 3037

Thu 30 Jan

Xhour lecture: Algebraic closure.

DF 13.4
FT 2425

Fri 31 Jan

Field automorphisms. Automorphism group.

DF 14.1
FT 3639

5

Mon 03 Feb

Constructing automorphisms. Automorphism group calculations.
Fixed fields.

DF 14.1
FT 3639

Problem Set #4

Wed 05 Feb

Galois extensions.
Linear independence of embeddings.

DF 14.114.2
FT 3637

Fri 07 Feb

Fundamental theorem of Galois theory.
Examples of the Galois correspondence.

DF 14.2
FT 3639

6

Mon 10 Feb

End of proof of the Galois correspondence.

DF 14.2
FT 3840

Problem Set #5

Wed 12 Feb

Normality.
Finite fields. Cyclotomic fields.

DF 14.23
FT 3748

Fri 14 Feb

Examples of Galois extensions. Computing Galois groups.

DF 14.2
FT 5254

7

Mon 17 Feb

Applications of the Galois correspondence.
Radical extensions. Solvability by radicals.

FT 4 pp. 4245

Takehome Midterm Exam

Wed 19 Feb

Radical extensions. Solvability by radicals.
Galois's solvability theorem.

DF 14.7
FT 4245, 7475

Fri 20 Feb

Solvability, continued.
Primitive element theorem.

DF 14.7
FT 5961, 7475

8

Mon 23 Feb

Discriminant. Galois perspective on quadratic and cubic extensions.
Quartic extensions.

DF 14.7
FT 4951

Problem Set #6

Wed 25 Feb

Quartic extensions.

DF 14.7
FT 5058

9

Mon 02 Mar

Finite fields. Cyclotomic fields.

DF 14.5
FT 6265

Problem Set #7

Wed 04 Mar

Infinite Galois theory.
Krull topology. Profinite groups.

DF 650652
FT 9196

Thu 05 Mar

Fundamental Theorem revisited.

DF 650652
FT 9196

Fri 06 Mar

TausskyTodd's theorem. Grace Hopper's thesis.

Notices article

10

Mon 09 Mar

Final Exam!


Final Exam Review
