# Galois Theory - Algebraic extensions

## Homework Statement

Let M/L and L/K be algebraic field extensions. Is M/K necessarily algebraic?

## Homework Equations

Tower law: [M:K]=[M:L][L:K]

## The Attempt at a Solution

If both M/L and M/K are finite extensions then by the tower law M/K is also a finite extension, hence is algebraic. So one or both of them must be infinite. The only infinite algebraic extensions I can think of are similar in construction to the algebraic closure of the rationals.
An element $$m \in M$$ is algebraic over L so we can write $$\sum a_{i}m^i=0$$ for some $$a_i \in L$$, where i runs from 0 to some n. Might be able to use the fact that L/K is algebraic now?

Thanks for any help!

## Answers and Replies

You have now singled out an element m and want to show that it's algebraic over K. Consider the tower:
$$K \subseteq K(a_1,\ldots,a_n) \subseteq K(a_1,\ldots,a_n,m)$$
You can show that m is algebraic over $K(a_1,\ldots,a_n)$ and you should therefore be able to get back to your finite case.

Funnily enough I was just working via this argument on paper to see whether I could delete the thread. At least now I know I was on the right lines, thanks very much!