Algebra Prelim

January 2006

1. Prove that the symmetric group S_{5} is generated by the two elements (1 2) and (1 2 3 4 5).

2. Let G be a group with |G| = 200. Show that G has a normal Sylow 5-subgroup which is abelian.

3. Let R be a ring (with identity) such that a^{2} = a for all a ∈ R. Prove that R is commutative.

4. Let Q be the field of rational numbers. Then:

(a) Argue that Q ⊂ Q(\sqrt{3}, \sqrt{2}) is a Galois extension and find its Galois group.

(b) Prove that Q(\sqrt{3}, \sqrt{2}) = Q(\sqrt{2 − \sqrt{3}}).

5. Determine all (up to isomorphism) rings S such that there is a surjective ring homonorphism R[X]/(X^{2} + 2)(X − 3)(X − 2) → S Here R is the field of real numbers.

6. Let k ⊂ L be a Galois extension where the degree of the extension is 17 · 11 · 7. Argue that for every intermediate field K (so k ⊂ K ⊂ L) K is a Galois extension of k.

7. Let M consist of all the n × n matrices A = (a_{ij}) over the field R of real numbers whose trace is 0 (i.e. ∑ a_{ii} = 0). Then:

(a) Argue that M is a vector space over R.

(b) Find the dimension of M.

(c) Decide if M is closed under the multiplication of matrices.

(d) If A, B ∈ M show that AB − BA cannot be the identity matrix.

8. Let f(X) = X^{12} −1 ∈ Q[X] (where Q is the field of rational numbers). Then:

(a) Find the Galois group of the splitting field E of f(X) over Q.

(b) Find all the subfields of E.

9. Let R = Z[\sqrt{−5}] and let K be the field of fractions of R. Show that the polynomial f(X) = 3X^{2} + 4X + 3 is irreducible in R[X] but that it is reducible in K[X].