Ramification in cyclotomic field I was trying to prove that:
$p$ is a odd prime and $\zeta$ is a primitive $p$-th root of unity and let $p^{*} = (-1)^{\frac{p-1}{2}}p$. Show that $\mathbb{Q}(\sqrt{p^{*}})$ is a quadratic subfield of $\mathbb{Q}(\zeta)$.
This is what I got so far:
The only prime ramified in $\mathbb{Q}(\zeta)$ is $p$, and the prime factorization in $\mathbb{Q}(\zeta)$ is: $(p) = (\mathcal{P_{1}}...\mathcal{P}_{n})^{\phi(p)=p-1}$ (Since it is Galois, so all primes have the same ramification degree). $p-1$is even, so I looked at $(\mathcal{P_{1}}...\mathcal{P}_{n})^{\frac{p-1}{2}}$. But I have trouble to determine what is this ideal exactly? Is this $(\sqrt{p^{*}})$? If it is, where does $(-1)^{\frac{p-1}{2}}$ comes from?
Any help is appreciated!

Let $p$ be an odd prime, and consider the cyclotomic field $K=\mathbf Q(\zeta_p)$, which is a cyclic extension of degree $(p-1)$. Since $p$ is odd,  by the Galois correspondence, $K$ contains a unique quadratic field $k=\mathbf Q (\sqrt d)$, and we want to show that actually $k=\mathbf Q (\sqrt {p^*})$ (in your notation). 
To this end, we can study the ramification of primes in $k$ and $K$ as you started to do. The quickest way is to use the discriminants of the rings of integers $O_k$ and $O_K$. It is classically known that $disc(O_K)=\pm p^{p-2}$ and $disc(O_k)= 4d$ if $d\equiv 2$ or $3$ mod $4$ , $d$ if $d\equiv 1$ mod $4$. It follows that $p$ is the only prime which ramifies in $K$ (hence the same is true for $k$), and the previous reminder on ramification in $k$ and the unicity of $k$ show that $k=\mathbf Q (\sqrt {p^*})$ . 
