# 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 $$\QQ$$?