Fall 2017 Problem 1 Suppose that F ⊆ K is an inclusion of fields and let α, β ∈ K be two elements which are algebraic over F. Show that α + β is also algebraic over F.

Fall 2013 Problem 6. Compute [ℚ(\sqrt{2}, \sqrt{3}) : ℚ] and find a basis for ℚ(\sqrt{2}, \sqrt{3}) over ℚ.

Fall 2016 Problem 5 Let F be a finite field and let L be a subfield of F generated by elements of the form x^{3} for all x ∈ F. Prove that if L ≠ F, the F has exactly 4 elements.

Fall 2013 Problem 5 Is it possible to have a field extension F ⊆ K with [K : F] = 2, where both fields F and K are isomorphic to the field ℚ(x)?

Fall 2017 Problem 2. Let f ∈ ℚ[x] be the minimal polynomial of 1+ \sqrt[3]{2}+ \sqrt[3]{4} over ℚ, and let K be the splitting field for f over ℚ. What is [K : ℚ] and what is Gal(K/ℚ)? (Note that you are not required to find f.)

Fall 2015 Problem 6. Find the ℚ dimension of the splitting field over ℚ of x^{5} − 3.

Spring 2010 Problem 6: Find an irreducible polynomial over ℚ for which the order of the Galois group is not the factorial of the degree of the polynomial.

Fall 2009 Problem 2: The field extensions ℚ(\sqrt{\sqrt{2}})/ℚ(\sqrt{2}) and ℚ(\sqrt{2})/ℚ are both Galois. Show that ℚ(\sqrt{\sqrt{2}})/ℚ is not Galois.

Fall 2004 Problem 6: Construct infinitely many nonisomorphic quadratic extensions of ℚ and use these to show that the Abelianization of Gal(\overline{ℚ}/ℚ) is not finitely generated.