1. ## Find trace

Let p be prime, and let $K = \mathbb {Q} [w]$, w is a primitive pth root of unity. Compute $Tr_{K}(w)$ and $Tr_{K} (1-w)$

Also, show that $(1-w)(1-w^2)... (1-w^{p-1}) = p$

proof.

So I have $w^p = 1$ with $w^{k} \neq 1 \ \ \ \forall k < p$

Let p be prime, and let $K = \mathbb {Q} [w]$, w is a primitive pth root of unity. Compute $Tr_{K}(w)$ and $Tr_{K} (1-w)$
If $E/F$ is a finite Galois extension then we define $\text{Tr}(\alpha) = \sum_{\theta \in \text{Gal}(E/F)}\theta (\alpha)$ for $\alpha \in E$. Now if $\omega$ is a primitive $p$-th root of unity then $\mathbb{Q}(\omega)$ is finite and Galois over $\mathbb{Q}$, furthermore, since $\{ 1,\omega, ... , \omega^{p-1}\}$ for a basis for this extension and $\mathbb{Q}$ is a field with $\text{Char}(\mathbb{Q}) = 0$ it means $G=\text{Gal}(K/\mathbb{Q})$ is a group of order $p$, so it must be a cyclic group. Thus, there is an automorphism $\theta$ of $K$ leaving $\mathbb{Q}$ fixed so that $\left< \theta \right> = G$. Suppose that $\theta (\omega) = \omega^k$ where $0< k \leq p-1$. Then it means $\{ \text{Id},\theta, \theta^2, ... ,\theta^{p-1} \} = G$. And so $\text{Tr}(\omega) = \text{Id}(\omega) + \theta(\omega) + \theta^2 (\omega) + ... + \theta^{p-1} (\omega)$. Thus, $\mbox{Tr}(\omega) = \omega + \omega^k + \omega^{2k} + ... + \omega^{(p-1)k}= \omega - 1$.