Sum of powers in finite fields I have trouble following the logic in this proof. In particular, why is the following equality is true:
$$\displaystyle\sum_{x \in K^\times} x^u = \displaystyle\sum_{x \in K^\times} y^u x^u$$

As $y \in K^{\times}$, the map $\varphi_y : K^{\times} \to K^{\times}$, $x \mapsto yx$ is a bijection (with inverse $\varphi_{y^{-1}}$). So the only difference between the two summations is the order of the summands, so they are equal.
For example, if $u = 1$, $K = \mathbb{Z}/5\mathbb{Z}$, and $y = 2$, then we have
\begin{align*}
\sum_{x\in K^{\times}}x &= 1 + 2 + 3 + 4\\
\sum_{x\in K^{\times}}2x &= 2 + 4 + 1 + 3.
\end{align*}
