Orderings of $\mathbb Q[\zeta]$

Question

I want to apply an Theorem, but for that I need to know, how many orderings the totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}[\zeta]$ has. I think possible answers are $1$ or $(p-1)/2$, but I don't know how to check this. Does anyone have an idea?

Best regards Laura

Answer 1

Hint: An embedding $K\to\mathbb{R}$ induces a total ordering on $K$ compatible with the field structure of $K$. How many distinct such embeddings are there?

Answer 2

Ah, I think I got it. The degree of the field extension is $(p-1)/2$ and the field is totally real, so there are $(p-1)/2$ embeddings and each embedding defines an ordering.

Thank you very much! Laura