Thread: Algebraic over K

Let K ⊆ E ⊆ F be fields. Prove that if F is algebraic over K, then F is algebraic over E and E is algebraic over K.

Here is what I am thinking:
Since F is algebraic over K, then F would be algebraic over E since E ⊆ F. Similarly, E would be algebraic over K since E ⊆ F and F is algebraic over K.

Does this make sense for a proof? If not, can you please help? Thanks in advance.

Originally Posted by page929

Let K ⊆ E ⊆ F be fields. Prove that if F is algebraic over K, then F is algebraic over E and E is algebraic over K.

Here is what I am thinking:
Since F is algebraic over K, then F would be algebraic over E since E ⊆ F. Similarly, E would be algebraic over K since E ⊆ F and F is algebraic over K.

Does this make sense for a proof? If not, can you please help? Thanks in advance.

No, that proves nothing. You must prove that $\displaystyle f\in F\Longrightarrow \exists 0\neq p(x)\in E[x]\,\,s.t.\,\,p(f)=0$ ,

But $\displaystyle f\in F\Longrightarrow \exists\,0\neq g(x)\in K[x]\,\,s.t.\,\,g(f)=0$ , and since

$\displaystyle K<E$ then also $\displaystyle g(x)\in E[x]\Longrightarrow f$ alg. over E.

Now you do the second one alone.

Tonio