2022 Spring Algebra

Problem 1.

If is a subgroup of a finite index in , show that contains a subgroup which is normal and of finite index in .

Problem 2.

Prove that there is no simple group of order 351.

Problem 3.

If , are normal subgroups of , show that is isomorphic to a subgroup of .

Problem 4.

Let be a field. For any nonzero ideal of , prove that there is an isomorphism of rings , where each has the property that its ideals all lie in a finite chain .

Problem 5.

Assume the following rings are commutative with multiplicative identity. For each of the following, either give an example (with explanation) of such an ideal or show why it does not exist:

(a) A prime ideal in a finite ring that is not maximal.

(b) A prime ideal in an integral domain that is non-zero but not maximal.

Problem 6.

Let , be two finite abelian groups of orders , , respectively, and assume that . Show that .

Problem 7.

Classify all modules over that have 16 elements. (Hint: They are also modules over .)

Problem 8.

Assume that an irreducible degree 7 polynomial over has a cyclic Galois group. Show that the order of this group must be 7.

Problem 9.

Let be a prime such that for a positive integer . Suppose that is an irreducible polynomial over . Show that its splitting field has degree over .

Problem 10.

Suppose that is a field extension with . If is a root of a quadratic polynomial in , show that .