Algebra Preliminary Exam January 1997

1. Prove that if A is an n × n matrix with coefficients in a field, then A is similar to a

matrix of the form ( \begin{matrix}A_{1} & & & & & 0 \\ & A_{2} \\ & & . \\ & & & . \\ & & & & . \\ 0 & & & & & A_{r} \end{matrix} ) where the characteristic polynomial of

A_{i} (i = 1 … r) is the power of an irreducible polynomial.

2. Prove that (p − 1)! ≡ −1 (mod p) for p an odd prime.

3. If G is any group and H is a subgroup of G with G : H = n, show that there exists a normal subgroup K of G such that K ⊆ H and G : K ≤ n!

4. Determine the structure of the Galois group G of the splitting field M over the rational numbers Q of the polynomial f(x) = x^{5} − 2. How many Sylow 2-subgroups does G have? Give the fixed subfields of M of each Sylow 2-subgroup. Do the same thing for the Sylow 5-subgroups. Which of these subfields are normal field extensions of Q?

5. Let K be a normal, separable extension field of F, and p(x) ∈ F [x] be an irreducible polynomial. If in K[x] p(x) = p_{1}(x)· . . . ·p_{r}(x) where p_{i}(x) are irreducible polynomials in K[x], i = 1 … r, prove that p_{1}(x), …, p_{r}(x) all have the same degree.

6. Let R be an integral domain. State and prove the universal mapping property for the embedding of R into its field of fractions.

7. Let R be a ring with unit, A, C right R-modules, B, D left R-modules, f : A → C a right R-module homomorphism, g : B → D a left R-module homomorphism. Let h : A ⊗_{R} B → C ⊗_{R} D be defined by h(a ⊗ b) = f(a) ⊗ g(b). If f and g are monomorphisms, is h necessarily a monomorphism? Why?

8. Let p be a prime number and Z_{p} be the completion of Z at the prime ideal pZ. Prove

that there exists a map

χ : F_{p} → Z_{p}

with the following properties:

(a) If π : Z_{p} → F_{p} is the canonical map, then

π · χ is the identity on F_{p}

(b) χ is multiplicative: that is, χ(ab) = χ(a) χ(b) for all a, b in F_{p}.