MMA310 H23 Galoisteori

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This page contains the program of the course lectures. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

Program

The schedule of the course is in TimeEdit.

Office hours / Fika Thursdays 14:30-15:30 at Café Bulten (M-huset)

 

Lectures

The course uses exercises as approach to Galois theory, with combined lectures and exercise sessions.

The program is preliminary.

 

Notes (last changes 09/01 - now including an appendix B with another example of an inverse Galois problem)

Whiteboard notes 1/11

 

Week Sections Content, recommended homework
44 2, 3

Solving algebraic equations, field extensions

2.1, 2.5, 2.7, 2.8; 2.9, 2.10, 2.11, 3.2, 3.6
45 3, 4, 5

Irreducibility of polynomials, algebraic extensions, splitting fields

3.8 (b),(c4-c6,) [4.4(b)&4.9]; 4.3 rem., 4.4 rem., 4.2, 5.1 rem., 5.4; 4.12, show that Qbar is countable.
46 5, 6, 7

Automorphism groups of fields, normal and separable extensions

5.3(+16.9), 5.8, 5.11; 5.5, 5.6, 5.7 (a1),(c); 6.1(b), 6.2, 6.3, 7.1 remaining, 7.3, diagram for Q(\sqrt2,\sqrt 3)/Q: which intermediate fields are invariant under which subgroups of the Galois group?
47 7, 8; 12

Normal and separable field extensions; Solvable groups

7.6, 7.7, 7.9; 8.1, 8.2 (rem.), 8.3(a); 12.1(b),(e), 12.6,
Challenge: Let G_0 = G and define G_i=[G_{i-1},G_{i-1}] as the commutator subgroup. This is the derived series. Show that G is solvable if and only if G_n={e} for some n.

For a reformulation of Ex.8.2, check the notes!
48 9, 10

Fundamental theorem of Galois theory

8.10 (a)-(e), 9.1, 9.2, 9.3, 9.4;
49 13

Solvability of equations

9.16 (b)-(h), 9.21; prove (F1) and (F2) from the notes
50 14

Geometric constructions, cyclotomic fields ( transcendence of e and LaTeX: \pi)

14.3,14.4,14.5; 10.1, 10.5(a); 10.7, 10.9, 14.7.
51 Course summary, exam preparation

 

Links

  • Biographies at the MacTutor History of Mathematics Archive.

 

 

Course summary:

Date Details Due