Ph.D. Preliminary Examination in Algebra

June 9, 2000

1. Let X be any set, F any field, and F^{X} the set of maps from X to F. F^{X} is endowed with the structure of a vector space over F using “pointwise” addition and multiplication by scalars. Prove that a finite sequence f_{1}, f_{2}, …, f_{n} of elements of F^{X} is linearly independent if and only if there is a finite sequence of elements x_{1}, x_{2}, …, x_{n} in X for which the n × n determinant det (f_{i}(x_{j})) is non-zero.

2. Find a complete set of representatives for the isomorphism classes of finite abelian groups of order 1001.

3. Let K be the splitting field over the field Q of rational numbers of the polynomial

f(x) = x^{5} − x^{4} + x^{3} − x^{2} + x − 1 .

(a) What are the possible values for the minimum degree among the irreducible factors of a polynomial of degree 5?

(b) Write f as the product of factors irreducible over R.

(c) Write f as the product of factors irreducible over Q.

(d) What is the degree of K over Q?

(e) What is the Galois group of K over Q?

4. Show that if two square matrices of the same finite size over a field are similar in a larger field then they must be similar in the original field.

5. Let F be a field, and let A be the quotient ring

A = F [t, x, y, z]/(tz − xy)F [t, x, y, z]

where t, x, y, z are independent transcendentals over F.

(a) Show that A has no zero divisors.

(b) Explain briefly why A is Noetherian.

(c) Is A a unique factorization domain? (Either prove that it is or exhibit an example of something

that does not factor uniquely according to the usual criteria for such uniqueness.)

6. Let E be a finite extension of a field F.

(a) Outline an argument for showing that if F is a finite field, then E is a cyclic Galois extension of F.

(b) Provide an example where F is a field of characteristic 5 and E is an extension of F of degree 5 that is not a Galois extension of F.

(c) For any given field K explain how to obtain an extension F of K and a finite extension E of F for which E is a Galois extension of F with Galois group isomorphic to the symmetric group S_{n} (consisting of the permutations of n objects).

7. Let F_{3} denote the field of 3 elements.

(a) What is the cardinality of 2-dimensional Cartesian space F_{3} × F_{3} over F_{3}?

(b) Let N denote cardinality of the group GL_{2}(F_{3}) of linear automorphisms of F_{3} × F_{3}. Compute N.

(c) Observe that the multiplicative group F_{3}^{∗} is the unique group of order 2 and furthermore that: i. Multiplication by invertible scalars gives rise to a homomorphism ϕ from F_{3}^{∗} to GL_{2}(F_{3}). ii. The determinant gives rise to a homomorphism ψ from GL_{2}(F_{3}) to F_{3}^{∗} Explain why the kernel of ψ and the cokernel of ϕ both have the same cardinality.

(d) Is the kernel of ψ isomorphic to the cokernel of ϕ?

8. Prove over any commutative ring (with 1) that two isomorphic free modules of finite rank must have the same rank.