2020 Spring Algebra

Problem 1.

Consdier the cube in the Euclidean vector space as the set of points for which the absolute value of each of the coordinates is less than or equal to 1. Let be the group of those rotations in preserving the origin which send to itself. Show that is isomorphic to the symmetric group .

Problem 2.

Let be a prime. Prove that if is a group of order then is not simple.

Problem 3.

Determine whether the quaternion group can be represented as a quotient of . Justify your answer.

Problem 4.

Show that is a Euclidean domain, with respect to the usual complex norm. Find an irreducible element of which is not in . Justify your answer.

Problem 5.

Let be a commutative ring with 1. Recall that an element of is idempotent if . Assume is finite and strictly more that elements of are idempotent. Let be the set of all idempotent elements of and .

(a) Prove that whenever .

(b) Prove that is a subgroup of of index at most 2.

(c) Prove that is a subgroup of and conclude that every element of is idempotent.

Problem 6.

We say that a finitely generated -module is invertible iff there exists a finitely generated -module such that .

(a) Find all invertible -modules.

(b) Prove that if and are both invertible then so is .

(c) Prove that every invertible -module is projective.

Problem 7.

Consider the polynomial .

(a) Find the splitting field of if we consider as a polynomial in .

(b) Find the splitting field of if we consider as a polynomial in .

Problem 8.

Let be a monic cubic polynomial with distinct roots . Let be the monic cubic polynomial with roots , , and .

(a) Prove that , that is, the coeficients of are in .

(b) Prove that if then .

Problem 9.

Let be a prime, be a finite field with elements and be a finite field with elements. Consider the map taking to its -th power . If is viewed as a vector space over , show that is -linear and find its characteristic and minimal polynomials.

Problem 10.

Prove that for every there exists an nonsingular matrix over such that is its inverse.