so if we're given that L extends K, w/ L algebraic of K, and M extends L, w/ M algebraic over L, do we get that M is algebraic of K? (we may not assume finite extensions)
that is:
For all l in L, there exists p(t) in K[t] such that p(l)=0
For all m in M, there exists q(t) in L[t] such that q(m)=0
For all m in M, does there exist r(t) in K[t] such that r(m)=0?
thanks for any help
Thanks!
I guess F:=K(a_0,\ldots,a_{n-1}) is a finite extension because, we have finitely many algebraic roots, each of which are the root of some minimal polynomial in K, so using the Tower Theorem, the degree is a product of a bunch of finite terms and thus finite. Then we simply use that finite \rightarrow algebraic (though the converse is false). So each m \in M is algebraic over K and M is algebraic over K.
Does that sound right?