Math 412, Fall 2018

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Math 412: Topics in Algebra, Fall 2018

MWF 1:40-2:30pm, Eliot 207

Office Hours: MWF 2:45-4:15pm in Lib 306

Text: Galois Theory (2nd ed.) by David Cox

412syllabus.pdf / 412syllabus.tex

Cumulative notes [pdf]

Week 1: August 27-31

M: §2.1 Polynomials of several variables. Syllabus. Notes.
W: §2.2 Symmetric polynomials. Notes.
F: §2.4 The discriminant. HW1 due. Notes.

Week 2: September 3-7

M: Labor Day holiday. No class.
W: §3.1. Existence of roots. Notes.
F: §3.2. The fundamental theorem of algebra. HW2 due. Notes.

Week 3: September 10-14

M: §4.1. Elements of extension fields. Notes.
W: §4.2. Irreducible polynomials. Notes.
F: §4.3. The degree of an extension. HW3 due. Notes.

Week 4: September 17-21

M: §4.4. Algebraic extensions. Notes.
W: §§5.1-5.2. Splitting fields and normal extensions. Notes.
F: §5.3. Separable extensions. HW4 due. Notes.

Week 5: September 24-28

M: §6.1. The Galois group. Notes.
W: §§6.2-6.3. Galois groups of splitting fields and permutation of the roots. Notes.
F: §6.4. Examples of Galois groups. HW5 due. Notes.

Week 6: October 1-5

M: §7.1. Galois extensions. Notes.
W: §7.2. Normal subgroups and normal extensions. Notes.
F: §7.3. The fundamental theorem of Galois theory. HW6 due. Notes.

Week 7: October 8-12

M: §7.3. More fundamental theorem of Galois theory. Notes.
W: Exam review. First exam distributed (two-hour take home, one two-sided page of notes, no other resources; covers up to and including Wednesday of Week 6).
F: §7.4. First applications of Galois theory. Exam 1 due at the start of class. Discuss final projects (assignment handout, paper template). Notes.

Fall break: October 15-19

Week 8: October 22-26

M: §§8.1-8.2. Solvable groups. Notes.
W: §8.2. Solvable extensions.
F: §§8.3. Solvable extensions and solvable groups. Notes. HW8 due.

Week 9: October 29 - November 2

M: §8.4. Simple groups. Notes.
W: §8.5. Solving polynomials by radicals. Notes.
F: §9.1. Cyclotomic polynomials. HW9 due. Notes.

Week 10: November 5-9

M: §10.1. Constructible numbers. Notes.
W: §10.2. Regular polygons and roots of unity. Notes.
F: §11.1. The structure of finite fields. HW10 due. Notes.

Week 11: November 12-16

M: §11.2. Irreducible polynomials over finite fields. Notes.
W: Introduction to ordered fields. Notes.
F: Real closed fields. HW11 due. First draft of final paper due. Notes.

Week 12: November 19-23

M: Exam review. Second exam distributed (two-hour take home, one two-sided page of notes, no other resources; covers Week 6 through Monday Week 11).
W: The Artin-Schreier theorem.
F: No class or HW. Thanksgiving holiday.

Week 13: November 26-30

M: Inverse Galois (JR), Transcendence of $e$ and $\pi$ (Zichen).
W: More constructibility (Alex), Origami (Pallavi).
Th (3:10-4:25pm in Lib203): Infinite Galois theory (Tristan), Hilbert's Satz 90 (Nick), Imprimitive solvable permutation groups (Miles). Also donuts 🍩.
F: Meet in P122. The Kronecker-Weber theorem (Caroline). Course evaluations.

Week 14: December 3-7

M: Geometry of automorphisms (Anton), the Galois correspondence in algebraic topology (Genya).
W: Polynomials of prime degree (Torin), Kronecker's construction of splitting fields (Livia).
F: No class or HW. Reading period.

Finals week: December 10-14

W: Final homework due by 9pm. [Assignment will be updated as the last few problems trickle in. See the Google Drive folder shared with you for copies of your peers' papers.]

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