Let $K\subset L\subset M$, where $L$ is an algebraic field extension of $K$, and $M$ is an algebraic field extension of $L$. Consider a $K$-homomorphism $\phi\colon L\to\overline K$, where $\overline K$ is an algebraic closure of $K$. I want to show that I can extend $\phi$ to some $K$-homomorphism $\tilde\phi\colon M\to\overline K$. One way of doing this is by looking at an $L$-homomorphism $\psi\colon L\to\overline K$. I was thinking of defining $$ \tilde\phi(x)=\begin{cases} \phi(x)&\text{if }x\notin L\\ \psi(x)&\text{if }x\in L. \end{cases} $$ But I'm not sure if this is a correct extension of a field homomorphism, and I'm not sure how to show that it is a homomorphism. If we have $x,y\in L$, then it's trivial that $\tilde\psi(xy)=\tilde\psi(x)\tilde\psi(y)$, but if $x\in L$ and $y\notin L$, I'm not sure what we would get.


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