Fall 2012 Problem 1:

(1) Can a vector space over an infinite field be a finite union

V = ∪_{i=1}^{k}V_{i}

where for each i, V_{i} ≠ V?

(2) Can the group ℤ^{2} = ℤ × ℤ be a union of fnitely many proper subgroups?

Spring 2013 Problem 1: Let A be a Boolean ring, i.e., a^{a} = a for all a ∈ A. Show that the ring A is commutative.

Fall 2012 Problem 4: Let R be a commutative ring and I an ideal in R. Prove or disprove: The set \sqrt{I} := \{a ∈ R : ∃n ∈ ℕ, n > 0, a^{n} ∈ I\} is an ideal.

Spring 2016 Problem 6: Find the smallest integer c ≥ 0 for which

R_{c} := ℤ[n]/(c, x^{2} − 2)

is (1) a domaiin

(2) a field.

Spring 2014 Problem 5. Let V be a linear representation of the group G over the field ℚ[\sqrt{2}], and let χ : G → ℚ[\sqrt{2}] be its character. Then V is also a vector space Vℚ over ℚ, and is again a linear representation of G. Express its character χℚ in term of the original character χ.

Spring 2015 Problem 3. Show that if G is an infinite simple group then every proper subgroup has infinitely many conjugates. Use this to conclude that G has an infinite automorphism group.

Spring 2012 Problem 5. Find the number of field homomorphisms ϕ : ℚ[2^{\frac{1}{3}} ] → ℂ.

Spring 2014 Problem 1. Find the smallest order of a group which is not cyclic and not isomorphic to a subgroup of the symmetric group Σ_{5}.