Mathematics 894 Problem Set 9

Text: Abstract Algebra, Third Edition, by Dummit and Foote

Suggested reading: Sections 13.1, 13.2, 13.4. We will omit section 13.3 in the lectures, although it is worth looking at to see a classical application of field theory.

Questions due next Friday (March 21, 08)

1. Page 519: 7,8,

2. Let α = \sqrt{3 + \sqrt{2}} and β = \sqrt{4 + 2\sqrt{3}}. Find [ℚ(α) : ℚ] and [ℚ(β) : ℚ]. Find the minimal polynomials of α and β over ℚ, and over ℚ(\sqrt{2}) (for α) and over ℚ(\sqrt{3}) (for β).

3. Page 545: 1-4

4. Let V = ℚ(i). As a two dimensional vector space over ℚ, V has basis {1, i}. Furthermore, multiplication by i is a ℚ-linear map from V to V. Find the matrix M of this linear transformation relative to the basis {1, i}. (You should be familiar with this from linear algebra, if not review page 415.) What are the eigenvalues of M? Find an eigenvector for each eigenvalue. Where do the eigenvectors live?