Spring 2014 Problem 2: Consider the four dimensional real vector space

V = {f : ℤ/4ℤ → ℝ}

with the ℝ[x]-module structure given by shifting so that (xf)(r) = f(r + 1). Find a direct sum decomposition of V into irreducible ℝ[x] modules.

Spring 2014 Problem 4: Give an example of a ring R and an R module which is projectve but not free.

Spring 2015 Problem 4: Find a quotient ring of ℤ[x] which is a pid but not a field.

Spring 2015 Problem 5: Let R = ℚ[X]/(X^{3} − 2).

(1) Is R a field? Explain.

(2) Run the extended Euclidean algorithm on X^{3} − 2 and X^{2} − X + 1 to find polynomials A(x) and B(x) with

A(X)(X^{3} − 2) + B(X)(X^{2} − X + 1) = gcd(X^{3} − 2, X^{2} − X + 1).

(3) Does [X^{2} − X + 1] have a multiplicative inverse in R? If yes, find it.

Fall 2018 Problem 5. Solve the following questions:

(a) If F is a field, prove that F [x]/(f(x)) is a field if and only if f(x) is irreducible over F.

(b) Show that f(x) = x^{2} + 2x + 2 is irreducible in ℚ[x], and find the inverse of 1 + x in ℚ[x]/(f(x)).

Fall 2017 Problem 4. Find all maximal ideals in the ring 𝔽_{7}[x]/(x^{2} + 1) and the ring 𝔽_{7}[x]/(x^{3} + 1).

Fall 2016 Problem 4. Let I, J ⊆ R be ideals in a principle ideal domain R. Prove that I + J = R iff IJ = I ∩ J.

Fall 2014 Problem 4. List all ideals of 𝔽_{p}[x]/(x^{2} + x − 6) when (a) p = 7

(b) p = 5.

Fall 2015 Problem 4. Let I denote the ideal in the ring ℤ[x] generated by 5 and x^{3} + x + 1. Is I a prime ideal?