I am given the p-cyclotomic field as $K_p = \frac{\mathbb{Q}[x]}{(\Phi_p(x))}$ where $\Phi_p(x) = x^{p-1} + x^{p-2} + ... + x + 1 $
Then, a quadratic subfield L is defined such that $\mathbb{Q} \subset L \subset K_p$ with $[L : \mathbb{Q}] = 2$.
The Gauss sum is then defined as:
$$\tau = \sum_{a\in (\mathbb{Z} / p \mathbb{Z})^\times} \left(\frac{a}{p}\right) \zeta^a$$
where the $\left(\frac{a}{p}\right)$ is the Legendre symbol.
I have shown that $\tau^2 = \left(\frac{-1}{p}\right) p$.
I am asked to show that this sum $\tau$ is contained in the quadratic subfield L. My main confusion comes from the concept of L. Not really sure what an element of L would look like. I have read about how L is simply $\mathbb{Q}$ extended with some squareroot of a square free number. However, I do not see why this number must be p.
In other words, why must $L = \mathbb{Q}(\sqrt p)$, and even if this is the case, how does this take into account for the fact that if $\left(\frac{-1}{p}\right) = -1$, then $\tau = i\sqrt p$.
Any help/info would be very much appreciated.