Algebra Prelim

June 1998

Provide proofs for all statements.

If necessary, you may use the results from other parts of this test, even though you may not have successfully proved them. All 10 questions carry equal weight in determining your grade.

1. Let F be a field and V a vector space over F. Let R = F ⊕ V be the ring whose operations are

defined by:

(f_{1}, v_{1}) + (f_{2}, v_{2}) = (f_{1} + f_{2}, v_{1} + v_{2})
(f_{1}, v_{1}) × (f_{2}, v_{2}) = (f_{1}f_{2}, f_{1}v_{2} + f_{2}v_{1})

(a) Argue that this makes R into a commutative ring with identity.

(b) Argue that (f, v) ∈ R is a unit if and only if 0 ≠ f ∈ F.

(c) Show that I ⊂ R is a proper ideal if and only if I = {(0, w)|w ∈ W } for some subspace W of V.

2. Find the irreducible monic polynomial satisfied by \sqrt{5} + \sqrt{3} over Q .

3. Show that a group G of order 28 always has a subgroup H of order 14.

Further show that there is at least one subgroup of order 14 which is abelian.

4. Suppose that a group of order 10 is acting on a set with 13 elements. Argue that G must fix at least one element of S.

5. Let R be an Euclidean ring such that for every nonzero irreducible p in R and positive integer t, the quotient ring R/(p^{t}R) is finite. Prove that for any nonzero q ∈ R the quotient ring R/(qR) is also finite.

Give an example of a ring R satisfying the above hypothesis.

6. Recall the definitions:

• The element p ∈ R is a prime element if pR is a prime ideal.

• The element q ∈ R is an irreducible element if ab = q implies that a or b is a unit.

Now answer the following:

(a) Give an example of a ring R with a prime ideal which is not principal.

(b) Give an example of a ring R with an irreducible element which is not prime.

7. Prove that a commutative ring in which all ideals are prime is either the zero ring or is a field.

8. Let Q denote the rational numbers and Q [x] the polynomials in one variable over Q . In Q [x]

one has

(x^{3} + x^{2} + x − 1)(2x + 3) + (x^{2} + x − 1)(−2x^{2} − 3x − 4) = 1.

Find T ∈ Q [x] such that (x^{3} + x^{2} + x − 1)T has remainder x + 5 when divided by x^{2} + x − 1. Justify your answer.

9. Let f_{1}(x) = x^{2} − 3x. Inductively define f_{n}(x) = ∫_{0}^{x} f_{n−1}(t)dt.

(a) Determine if the ideal I generated by {f_{1}, f_{2}, ⋯, f_{n}, ⋯} in Q [x] is finitely generated.

(b) If the ideal I is finitely generated, then find an explicit set of generators with the least number of elements. Justify your answer.

10. Let F be a field of characteristic 3 and let K = F (t) where t is an indeterminate over F. Let L = F (t^{3} − t), the field generated by t^{3} − t over F.

(a) Argue that the map σ defined by σ(h(t)) = h(t + 1) is an L-automorphism of K.

(b) Given a rational function h(t) ∈ F (t), prove that h(t) ∈ L only if σ(h(t)) = h(t).

(c) Let h(t) be in K but not in L. Prove that the expression

(y − h(t))(y − h(t + 1))(y − h(t + 2))

is an irreducible polynomial in L[y].