Exercise 18, § 11

Let $p \in \mathbf{N}$ be a prime number, $C_p$ the cyclotomic extension of level $p$ of $\mathbf{Q}$, $\zeta \in C_p$ a primitive p-th root of unity and $n$ an integer. Write

a) Calculate $d = [\mathbf{Q}(X_n(\zeta)) : \mathbf{Q}]$. Determine the Galois group of the extension $\mathbf{Q}(X_n(\zeta))$ of $\mathbf{Q}$ and the conjugates of $X_n(\zeta)$. b) Let $T^d + a_1 T^{d-1} + \ldots + a_d$ be the minimal polynomial of $X_n(\zeta)$ over $\mathbf{Q}$. Show that $a_1 = 0$. c) Suppose that $n > 1$ and $p \equiv 1 \pmod{n}$. Show that $a_1 = -\frac{n(n-1)p}{2}$ if $\frac{p-1}{n}$ is even and $a_1 = \frac{np}{2}$ if $\frac{p-1}{n}$ is odd. d) Suppose that $p \equiv 1 \pmod{n}$. Let $\beta \in \mathbf{F}_p^*$ be an element whose residue class in $\mathbf{F}_p^*/\mathbf{F}_p^{*n}$ is a generator. Let $\mu$ be the number of solutions in $\mathbf{F}_p^n$ of the equation $\sum_{i=1}^n \beta^{i-1} X_i^n = 0$. Show that $a_1 = \frac{(-1)^n p (\mu - p^{n-1})}{p-1}$ e) Calculate $a_1$ when $n = 3$ and $p = 7, 13, 19$.

\* ^1 The largest totally ramified extension of the field $\mathbf{Q}_p$ of p-adic numbers in its algebraic closure is a field of characteristic 0 with the same property. \*