Field embedding

Let $K$ and $L$ be fields . A field embedding $\sigma:K\hookrightarrow L$ is a ring homomorphism $\sigma:K\to L$ such that $\sigma(1)=1$ and $\sigma$ is injective.

More generally, given a field extension $L/K$ and a field $\Omega$ containing $K$, a $K$-embedding of $L$ into $\Omega$ is a field embedding $\sigma:L\hookrightarrow \Omega$ whose restriction to $K$ is the identity map. In this setting, $\sigma$ is also called a $K$-homomorphism. If $\sigma$ is bijective, it is a field automorphism of $L$ (and if it fixes $K$, a $K$-automorphism).

Field embeddings are the basic inputs for expressing the trace and norm as sums/products of conjugates when the extension is finite and separable.

Examples

1. Inclusion map. If $K\subseteq L$ (so $L/K$ is a field extension ), the inclusion $K\hookrightarrow L$ is a field embedding.

2. Two $\mathbb{Q}$-embeddings of a quadratic field. Let $L=\mathbb{Q}(\sqrt{d})$ with $d$ squarefree. Then there are two $\mathbb{Q}$-embeddings $L\hookrightarrow \mathbb{C}$: one sends $\sqrt{d}\mapsto \sqrt{d}$, the other sends $\sqrt{d}\mapsto -\sqrt{d}$.

3. Embeddings of a simple extension from roots. Let $L=\mathbb{Q}(\alpha)$ be a simple extension with $\alpha$ algebraic, and let $m_\alpha(x)\in\mathbb{Q}[x]$ be the minimal polynomial. Any $\mathbb{Q}$-embedding $\sigma:L\hookrightarrow \mathbb{C}$ is determined by the choice of a root $\beta$ of $m_\alpha$, via $\sigma(\alpha)=\beta$.