Question 15

Let \(p\) be a prime number. Let \(A\) be a \(p \times p\) matrix over a field \(F\) with 1 in all entries except 0 on the main diagonal.

Determine the Jordan canonical form (JCF) of \(A\)

  1. When \(F = \QQ\),

  2. When \(F = \FF_p\).

Hint: In both cases, all eigenvalues lie in the ground field. In each case find a matrix \(P\) such that \(P\inv AP\) is in JCF.

Question 16

Let \(\zeta = e^{2\pi i/8}\).

  1. What is the degree of \(\QQ(\zeta)/\QQ\)?

  2. How many quadratic subfields of \(\QQ(\zeta)\) are there?

  3. What is the degree of \(\QQ(\zeta, \sqrt[4] 2)\) over \(\QQ\)?