# $\sigma \in \mathrm{Gal}(K/k), \sigma \alpha \ne \alpha$, but why is $\alpha \in k$?

Suppose that $$k$$ contains $$\zeta$$, a primitive $$p$$-th root of unity where $$p$$ is prime, and that $$K$$ is Galois over $$k$$ with $$[K : k]=p$$; and write $$G=\operatorname{Gal}(K / k) \approx C_p$$. Show that $$K=k\left(\sqrt[p]{\alpha}\right)$$ for some $$\alpha \in k$$.

Hint: Let $$\sigma$$ be a generator of $$G$$. Take $$\alpha=\sum_0^{p-1} \zeta^{\nu} \cdot \sigma^{\nu} \beta$$ for $$\beta \in K$$ and show that one can choose $$\beta$$ so that $$\alpha \neq 0$$.

But with a simple try, I find that $$\sigma \alpha=\sum_{0}^{p-1} \zeta^{\nu} \cdot \sigma^{\nu+1} \beta\neq\alpha,$$ so why is $$\alpha$$ in k?

• Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Apr 15, 2019 at 14:16
• Here's what I see. Apr 15, 2019 at 14:30
• @Shaun I tried to edit Apr 15, 2019 at 14:37

This is just conflicting notation. The $$\alpha$$ in the hint is not the $$\alpha$$ in the assertion.
Show that you can find $$\beta\in K$$ such that $$\gamma:= \sum_{\nu=0}^{p-1}\zeta^\nu\cdot \sigma^\nu(\beta)\neq 0$$. Observe that $$\sigma(\gamma) = \zeta^{-1}\gamma$$ and conclude $$\gamma\in K\setminus k$$ and $$\alpha:=\gamma^p \in k$$.