2017 Spring Algebra

Problem 1.

Suppose is a group of order 80. Prove that is not simple.

Problem 2.

Prove that the additive group is isomorphic to the multiplicative group .

Problem 3.

Let be an integral domain. A nonzero nonunit element is prime if implies that or . A nonzero nonunit is irreducible if implies that or is a unit.

(a) Show that every prime element is irreducible.

(b) Show that if is a Unique Factorization Domain (UFD), then every irreducible element is prime.

Problem 4.

Let and be finite abelian groups, and suppose that the order of is relatively prime to the order of . Show that .

Problem 5.

Let be the dihedral group of order 8.

(a) Compute the center of .

(b) Compute the commutator subgroup of .

(c) Compute the conjugacy classes of .

Problem 6.

Let be a commutative ring with 1, and let be an -module. Show that if is a finitely generated -module, then is a finitely generated -module.

Problem 7.

(a) Find a polynomial whose splitting field has Galois group isomorphic to .

(b) Find a polynomial whose splitting field has Galois group isomorphic to .

(c) Find a polynomial whose splitting field has Galois group isomorphic to .

Justify your answers.

Problem 8.

Suppose is a perfect field and is a nonconstant polynomial. Show that is a direct product of fields if and only if is a separable polynomial.

Note: A previous version of this problem incorrectly left out the assumption that is perfect.

Problem 9.

Suppose is a prime.

(a) Show that all matrices of order exactly have the same characteristic polynomial, and find that polynomial.

(b) Show that all matrices of order exactly have the same minimal polynomial, and find that polynomial.

Problem 10.

Let and let .

(a) Show that is the splitting field of .

(b) Find a generator of the multiplicative group .

(c) Express the roots of in terms of .