2021 Winter Algebra

Problem 1.

Assume is a finite group and is a normal subgroup of . Recall that is the number of Sylow -subgroups of . Prove that .

Proof.

Problem 2.

Decide: Is it possible for the symmetric group to act transitively on a set of cardinality 14? Provide a proof to support your claim.

Proof.

Problem 3.

Let be a principal ideal domain. Suppose that are such that which and are relatively prime. Prove that

Proof.

Problem 4.

Consider an integral domain and its corresponding field of fractions . Assume is a monic polynomial and that it is possible to write as a product

where are monic polynomials of degree smaller that and at least one of , is not in . Prove that is not a unique factorization domain.

Proof.

Problem 5.

Let be a commutative ring and let . For an -module , denote , which is a submodule of . If

is an exact sequence of -modules and , prove that

is also an exact sequence.

Proof.

Problem 6.

Consider an integral domain and a principal ideal of , whihc is viewed as an -module. Let be the -module . Prove that the only torsion element of is zero.

Proof.

Problem 7.

Recall that denote the finite field with elements. Find a polynomial such that . Prove that your polynomial yields the desired isomorphism.

Proof.

Problem 8.

Consdier a subfield of the field of real numbers . Let and where is an -th root of in and is odd. Assume is a Galoid extension of such that . Prove that .

Remark: The following theorems from the course may be useful:

(A) Assume is a field of characteristic 0 which contains all -th roots of unity. Then the followng holds: If and is an -th root of then the extension is cyclic of order dividing .

(B) Assume is a Galois extension of fields and is any finite extension of fields. Then is a Galois extension and

Proof.

Problem 9.

Assume is a matrix over the complex numbers such that all eigenvalues of are non-zero. Prove that has a square root in , that is, there is a matrix such that .

Remark: It may be helpful to examine the Jordan form of the square of a Jordan cell.

Proof.

Problem 10.

Find a non-singular matrix of smallest possible dimension such that is its own inverse and is not a scalar multiple of .