# field embeddings?

I'm reading Algebraic Number Theory by Neukirch and unfortunately I have a weak background in field theory. In section 5, we consider an algebraic number field $K| \mathbb{Q}$ of degree n. This gives us a canonical mapping

$j:K \rightarrow K_\mathbb{C} := \prod _\tau \mathbb{C} ,$ where $a \mapsto ja=\tau a$

which comes from the $n$ different complex embeddings $\tau: K \rightarrow \mathbb{C}.$

What exactly are the $n$ embeddings $\tau: K \rightarrow \mathbb{C}$?

## 1 Answer

I expect that this comes from writing $K=\mathbb Q(\alpha)$ using the primitive element theorem.

Then the $n$ embeddings $\tau: K \rightarrow \mathbb{C}$ correspond to the $n$ complex conjugates of $\alpha$.

• Ok, so we just multiply an element of $K$ with a complex conjugate of $\alpha$? – 010110111 Oct 5 '15 at 18:49
• @010110111, no, the notation $\tau a$ means $\tau(a)$, the application of $\tau$ to an element $a$. More precisely, each element in $K$ is a polynomial expression $p(\alpha)$; an embedding $\tau$ sends $p(\alpha)$ to $p(\beta)$, where $\beta$ is a complex conjugate of $\alpha$. – lhf Oct 5 '15 at 19:09