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## 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 [tex]m \in M[/tex] is algebraic over L so we can write [tex]\sum a_{i}m^i=0[/tex] for some [tex]a_i \in L[/tex], 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!