Field homomorphism

A field homomorphism preserves addition, multiplication, and the multiplicative identity.

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

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.