Algebra Preliminary Exam
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
where the characteristic polynomial of
Ai (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) = x5 − 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) = p1(x)· . . . ·pr(x) where pi(x) are irreducible polynomials
in K[x], i = 1 . . . r, prove that p1(x), . . . , pr(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 Zp be the completion of Z at the prime ideal pZ. Prove
that there exists a map
χ : Fp → Zp
with the following properties:
(a) If π : Zp → Fp is the canonical map, then
π · χ is the identity on Fp
(b) χ is multiplicative: that is, χ(ab) = χ(a) χ(b) for all a, b in Fp.