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Math Help - Algebraic over K

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by page929 View Post
    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 f\in F\Longrightarrow \exists 0\neq p(x)\in E[x]\,\,s.t.\,\,p(f)=0 ,

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

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

    Now you do the second one alone.

    Tonio
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