2019 Spring Algebra

Problem 1.

Does the symmetric group contain a subgroup isomorphic to:

(a) The dihedral group with 8 elements?

(b) The quaternion group with 8 elements?

Problem 2.

Supppose is a finitely generated abelian group, a subgroup of , and . Prove that if is torsion free then the isomorphism classes of and determine the isomorphism class of uniquely. Give a counterexample that shows the isomorphism class of may not be uniquely determined if has a non-trivial torsion.

Problem 3.

For a group , let and let . We say is nilpotent if for some . Prove that if is a -group, i.e. for some prime , then is nilpotent.

(Recall that if , are subgroups of then where .)

Problem 4.

Throughout this question we assume that is a commutative ring with 1.

(a) Let be a multiplicative subset of (that is, and whenever ). Consider an ideal of such that , and is maxiimal with this property (that is, whenever is an ideal of ). Prove that is a prime ideal of .

(b) Recall that

is an ideal of , called the nilradical of . (Do not prove that is an ideal!). Prove that the following are equivalent:

(i) has exactly one prime ideal.

(ii) Every element of is either nilpotent (that is, an element of ) or a unit.

(iii) is a field.

Problem 5.

Recall that the ring of Gaussian integers has a Euclidean norm.

(a) Prove that for every ideal of , the quotient is a finite ring.

(b) Identify what is .

Problem 6.

Assume is a squarefree integer, i.e. is a product of distinct primes. Prove that the primitive -th roots of unity constitute a basis of the cyclotomic field over . (Here "basis" is meant in the sense of vector spaces.)

Problem 7.

Calculate the number of primitive elements of over . Recall that if is a field extension then is called a primitive element of over if and only if .

Problem 8.

Let be a Galois algebraic extension with not proper intermediate fields. Prove that is prime.

Problem 9.

Let be a vector space over the field of rational numbers of dimension at most where is a prime. Let be a linear operator on such that . Show that .

Problem 10.

Consider matrices , over such that the following are satisfied:

(i) , are nilpotent with the same nilpotency index (recall the nilpotency index of a matrix is the smallest number such that ).

(ii)

(iii)

Prove the following:

(a) If then , may be non-similar.

(b) If then , are similar.