# [SOLVED] Transitivity of Algebraic extension Fields

• Feb 19th 2010, 05:10 AM
verdi
[SOLVED] Transitivity of Algebraic extension Fields
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
• Feb 19th 2010, 07:49 AM
tonio
Quote:

Originally Posted by verdi
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

Let $\displaystyle m\in M\Longrightarrow \,\,\exists \,0\neq p(x)=x^n + a_{n-1}x^{n-1}+\ldots a_1x+a_0\in L[x]$ s.t. $\displaystyle p(m)=0$.
Now look at $\displaystyle F:=K(a_0,\ldots,a_{n-1})$ . This is a finite extension of $\displaystyle K$ ( why?) and $\displaystyle m$ is algebraic over$\displaystyle F$ , so it is algebraic over $\displaystyle K$ (why?) .

Tonio
• Feb 19th 2010, 11:54 AM
verdi
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?
• Feb 19th 2010, 01:59 PM
tonio
Quote:

Originally Posted by verdi
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?

Sounds right and dandy. Good! (Wink)

Tonio