2018 Spring Algebra

Problem 1.

Classify groups of order . Justify your answer. You can assume that 1009 is a prime number.

Problem 2.

Let P|P|=p^rp$.

(a) Prove that .

(b) Prove that is solvable.

Problem 3.

Let be a maximal ideal. Prove that is a finite field.

Problem 4.

Let be a UFD and assume that any ideal in is finitely generated. Suppose that for every nonzero , and any in is expressible as for some . Prove that is a PID.

Problem 5.

Classify all finite abelian groups such that .

Problem 6.

Let be a field and let nad be non-singular matrices over . Suppose that .

(a) Find the characteristic of .

(b) If is a positive or negative integer not divisible by 3, prove that the matrix has trace 0.

(c) Prove that the characteristic polynomial of is for some .

Problem 7.

Let be a field, and let be an matrix over . Suppose that is an irreducible polynomial such that . Show that .

Problem 8.

Let be a field and let be an irreducible polynomial. Suppose is a splitting field for over and assume that there exists an element such that both and are roots of .

(a) Show that the characteristic of is not zero.

(b) Prove that there exists a field between and such that the degree is equal to the characteristic of .

Problem 9.

Let be a finite field and let . Let be a splitting field over of . Prove that .

Problem 10.

For the alternating group ,

(a) Classify the conjugacy classes of .

(b) Construct the character table of .