I am trying to construct an embedding $\varphi:\mathbb{Q}(\zeta_{p})\to\mathbb{C}$, where $\zeta_{p}$ is the $p$-th root of unity. I then need to describe complex conjugation as an element of $\text{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ using the inclusion $\varphi$. I don't even know where to start.
My specific example is where $p=5$, but the general case would be very helpful.
Well, say your field is $K=\Bbb Q(\zeta)$, where $\zeta$ is a chosen primitive $n$-th root of unity, maybe if $n=5$ you choose $\cos(2\pi/5)+i\sin(2\pi/5)$. Then if you’re embedding into $\Bbb C$, you must sent $\zeta$ to a primitive $n$-th root of unity: if $n=5$, there are four of these; in general, there are $\phi(n)$ primitive $n$-th roots of unity. And you check that every such root determines the embedding completely. Finally, you check that sending $\zeta$ to $1/\zeta$ determines an embedding that agrees with complex conjugation on $K$. (Only on $K$, mind you: this embedding is not defined on all of $\Bbb C$, so it’s not quite right to ask for your embedding to “be” complex conjugation.)