$p$ is an odd prime, $z$ is a primitive $p$th root of 1. Prove that $\mathbb{Q}(z)$ contains the field $\mathbb{Q}(\sqrt{(-1)^{\frac {p-1}{2}}p})$

Asked Oct 02 '17 at 07:20

Active Oct 02 '17 at 07:20

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I was hoping someone could help me solve the following exercise:

Let $p$ be an odd prime, and let $z$ be a primitive $p$th root of 1. Prove that $\mathbb{Q}(z)$ contains the field $\mathbb{Q}(\sqrt{(-1)^{\frac {p-1}{2}}p})$

Thanks in advance, any suggestions would be greatly appreciated.

asked Oct 02 '17 at 07:20

Brassican

2

You can use the fact that the quadratic subfield only ramifies at $p$, or quadratic Gauss sums, or search this website for an answer. – Oct 02 '17 at 07:23
1

Gauss sums – Jyrki Lahtonen Oct 02 '17 at 07:37
How were you "hoping someone could help you" if you give zero indication about your background and how you tried to solve this? After 20+ months on the site, you are still not following the drill? – Did Oct 02 '17 at 07:41
I'm self studying fields and Galois theory, and I'm quite stuck on this question. – Brassican Oct 02 '17 at 07:47
In that case I very much recommend that you look up Gauss sums. Here's the case $p=7$ spelled out – Jyrki Lahtonen Oct 02 '17 at 08:01

$p$ is an odd prime, $z$ is a primitive $p$th root of 1. Prove that $\mathbb{Q}(z)$ contains the field $\mathbb{Q}(\sqrt{(-1)^{\frac {p-1}{2}}p})$

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