MASTER’S EXAM, ALGEBRA, JUNE, 1993

NOTATION: If f : X → Y is a map of sets, A ⊆ X, B ⊆ Y, then f(A) = {f(a) | a ∈ A} and f^{−1}(B) = {x ∈ X | f(x) ∈ B}. For n a positive integer, Z_{n} denotes the group (or ring) of integers modulo n.

1. Construct a nonabelian group of order 55 (Hint: Use the semi-direct product).

2. Find the elementary divisors and invariant factors of Z_{4} ⊕ Z_{15} ⊕ Z_{36} ⊕ Z_{100}.

3. Let G be a group of order 175 = 5^{2} · 7. Show that the 7-Sylow subgroup is normal and lies in the center of G.

4. Let f : V → W be a linear transformation of vector spaces and let K be the kernel of f.

(a) If U is a subspace of V, show that f^{−1}(f(U)) = U + K.

(b) If X is a subspace of W, show that f(f^{−1}(X)) = X ∩ f(V).

5. Find all maximal ideals and all nilpotent elements of the ring Z_{36}. (An element x of a ring R is called \underline{nilpotent} if x^{n} = 0 for some positive integer n.

6. (a) Find the Jordan canonical form of the matrix

( \begin{matrix}0 & 0 & 0 & −1 \\ 1 & 0 & 0 & −4 \\ 0 & 1 & 0 & −6 \\ 0 & 0 & 1 & −4 \end{matrix} ) .

(b) Find the rational canonical form of the matrix

( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix} ) .

7. (a) Show that [Q(\sqrt{2}, \sqrt{3}) : Q] = 4 and find a basis for this extension.

(b) Show that [Q(\sqrt{2} + \sqrt{3}) : Q] = 4 and find the minimum polynomial of \sqrt{2} + \sqrt{3} over Q.

8. Find the Galois group of the polynomial x^{4} − 4 over Q.