Field embedding

A field embedding $L\to \Omega$ is an injective field homomorphism. A $K$-embedding fixes every element of the base field $K$.

An embedding of a subfield into an algebraically closed field extends over an algebraic extension: for a simple extension, map its generator to a root of the transported minimal polynomial, then iterate or use Zorn lemma. If the larger extension is normal over the fixed base, its image under the extended embedding is itself.

Distinct field embeddings ${\sigma }_1,\dots ,{\sigma }_n:L\to \Omega$ are linearly independent over $\Omega$ as functions on $L$. Thus a nontrivial linear combination ${\sum }_i{a}_i{\sigma }_i$ cannot vanish on every element of $L$.

For the proof, choose a vanishing relation with the fewest nonzero coefficients and normalize one coefficient to one. If ${\sigma }_1\ne {\sigma }_m$, choose $y$ with ${\sigma }_1(y)\ne {\sigma }_m(y)$. Evaluating at $xy$ and subtracting ${\sigma }_m(y)$ times the relation evaluated at $x$ eliminates its $m$th term but leaves a nonzero first coefficient, contradicting minimality.

If $K/k$ is finite, then any distinct $k$-automorphisms ${\sigma }_1,\dots ,{\sigma }_m$ of $K$ are linearly independent in the $K$-vector space ${\mathrm{Hom}}_k(K,K)$, whose dimension over $K$ is $[K:k]$. Hence $m\le [K:k]$. In particular, a finite automorphism group $G$ satisfies $|G|\le [K:K^G]$.

If $\sigma$ has order $n$ and the fixed field contains a primitive $n$th root $\zeta$, thensatisfies $\sigma (T_{\zeta }(\beta ))=\zeta T_{\zeta }(\beta )$. Independence of the powers of $\sigma$ ensures that $T_{\zeta }(\beta )$ is nonzero for some $\beta$.

An embedding of $K({\alpha }_1,\dots ,{\alpha }_i)$ extends across ${\alpha }_{i+1}$ by sending it to a root of its transformed minimal polynomial. There are between one and the relative degree many choices, with equality for a separable extension.