MATH 4010/6010, Exam 2 Review

The exam Thursday 4/9 will cover the proof that the group Icosa is isomorphic to the group A₅, and the proofs that that these groups are simple, as well as sections 5.1, 7.3, 7.5, and half of section 7.6.

The following topics will NOT be on the exam: Lemma 7.6.3, Proposition 7.6.4, Proposition 7.6.6, Corollary 7.6.7, Theorem 7.6.11 and everything on pages 270-276.

The general suggestions for exam preparation are the same as for exam 1:

Make a vocabulary list. You should know the definitions and examples of the concepts defined.

Make a theorem list. You should know the statements of the theorems and examples illustrating the theorems.

Make an example list. In particular you should have a good list of field extensions and their Galois groups.

Review all the assigned homework. It's a good idea to look at all of the problems in the book, especially the low-numbered problems.

Again, questions on the exam will be of two basic types: short proofs and working out relatively simple examples. The proofs and examples may overlap what we've done in class and homework, or they may be new.

Key theorems to be covered on the exam:

A₅ is a simple group.

The icosahedral group is isomorphic to A₅. You do not have to know the proof, since it was a challenge problem.

Burnside's Theorem (7.3.1).

Characterization of direct product of groups (7.5.3).

Sylow Theorems (7.5.6) (7.5.8).

Elements of the Galois group G(K/F) permute the roots in K of a polynomial with coefficients in F. If K is the splitting field of a polynomial with coefficients in F, and this polynomial has no repeated roots in K, then G(K/F) is a subgroup of the group of permutations of the roots (7.6.2).

G(K/F) ≤ [K:F] (7.6.5).

If K is a Galois extension of F and g(x) is an irreducible polynomial in F[x] with a root α in K, then g(x) splits in K (7.6.9).