$L\to\overline{L}$ $K$-homomorphism restricts to an automorphism of L if L a splitting field of K

Question

Let $L$ be a splitting field of a family of polynomials in $K[X]$. Then, I am trying to prove that every $K$ homomorphism $\sigma\colon L\to\overline{L}$ restricts to an automorphism of $L$. Here $\overline{L}$ is the algebraic closure of $L$. What I can see is that $\sigma(L)$ is also a splitting field of the same family, hence we know that $\sigma(L)\cong L$ for two splitting fields, but is far from $\sigma(L)=L$, since as far as I am concerned, the isomorphism I am providing is non-canonical. What can be done?

Answer 1

For simplicity let $f\in K[X]$ and let $L$ be the splitting field of $f$, i.e. in $L$ we have a factorization $$f=(X-a_1)\cdots (X-a_n), a_i\in L$$ and $L=K(a_1,\ldots,a_n)$. Now take some $K$-homomorphism $\sigma : L\to \overline{L}$ then for $a_i,i=1,\ldots,n$ we have: $$ f(\sigma(a_i))=\sigma(f(a_i)) = 0 $$ by $\sigma|_K=id$. So $\sigma(a_i)$ is a zero of $f$. Thus, $\sigma(a_i)\in L$ for all $i$ and hence $\sigma(L)\subset L$. As $\sigma$ is injective it induces a bijection on the zero set of $f$, so we get $L\subset \sigma(L)$.