2020 Fall Algebra

Problem 1.

Consider . Prove that there are permutations both of order 2 such that has order .

Problem 2.

Give an example of a semi-direct product product of two abelian groups which is not abelian. Justify your example by an explanation why it works.

Problem 3.

Let be hte field of fractions of the integral domain , which is called the field of rational functions. For the subring

of , prove the following:

(a) is a principal ideal domain.

(b) has a unique irreducible element up to associates.

Problem 4.

Consider the ideal of the polynomial ring which is generated by a prime number and a non-constant polynomial . Prove that is maximal if and only if is irreducible modulo .

Problem 5.

Suppose that is a field of characteristic 5. For which values of is the polynomial separable?

Problem 6.

Let be a prime power. Consider the finite field as an abelian group under addition. For which is this group cyclic?

Problem 7.

Let be a field extension of degree 5 and the smallest subfield in the algebraic closure of , such that is Galois over and contains . Show that the degree of over is at most 120.

Problem 8.

Give an example of an injective map of abelian group , and a abelian group , such that is not injective. (Here is the tensor product over the ring of integers.) Justify your example by an explanation why it works.

Problem 9.

For a matrix , prove that the following are equivalent:

(a) The only eigen value of is .

(b) There exists such that is the zero matrix.

Problem 10.

Suppose that is a linear operator on a finite dimensional vector space over the field of rational numbers, and that has characteristic polynomial which is irreducible over . Show that the matrix of (in any basis of ) can be diagonalized over the field of complex numbers.