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Thread: Field Extensions: Algebraic and Transcendental Elements

  1. #1
    Senior Member roninpro's Avatar
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    Field Extensions: Algebraic and Transcendental Elements

    Hello there. I have the following problem:

    Suppose that $\displaystyle v$ is algebraic over $\displaystyle K(u)$ for some $\displaystyle u$ in an extension field of the field $\displaystyle K$. If $\displaystyle v$ is transcendental over $\displaystyle K$, show that $\displaystyle u$ is algebraic over $\displaystyle K(v)$.

    I've tried writing down the minimal polynomial for $\displaystyle v$ over $\displaystyle K(u)$ and playing around with that, but I haven't had any luck. I am also somewhat baffled by the transcendental assumption - where does it even come in to play, and how can I write it down? (I usually see it defined as a negative.)

    Just a hint in the correct direction would be appreciated.
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  2. #2
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    Hi

    Suppose that is algebraic over for some in an extension field of the field . If is transcendental over
    I just write what the hypotheses mean:

    $\displaystyle \mu_{P,K(u)}\in K(u)[X]-K[X]$ . I think it is sufficient to conclude
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  3. #3
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    Not sure if this is right but have a look at it see what you think;

    Let v be algebraic so that there is $\displaystyle f(x) \in K(u)[x]$
    with $\displaystyle f(v)=\sum_{i=0}^ma_iv^i=0$ and $\displaystyle a_i \in K(u)$.
    let v be transcendental over K so that $\displaystyle a_i=u^n$ for some n>0.

    so $\displaystyle f(x)=\sum_{j=0}^na_{i_j}x^{i_j}+\sum_{j=n+1}^mu^{i _j}x^{i_j}$
    $\displaystyle a_{i_j} \in K$

    then
    $\displaystyle f(v)=\sum_{j=0}^na_{i_j}v^{i_j}+\sum_{j=n+1}^mu^{i _j}v^{i_j}=\sum_{j=0}^m(a_{i_j}v^{i_j}+v^{i_j})u^{ i_j}=0$

    and $\displaystyle a_{i_j},v^{i_j} \in K(v)$ so $\displaystyle a_{i_j}v^{i_j} \in K(v)$

    so letting

    $\displaystyle g(x)=\sum_{j=0}^mc_jx^{i_j}$

    where

    $\displaystyle c_j=a_{i_j}v^{i_j}+v^{i_j} \in K(v)[x]$

    and g(u)=0 so v is algebraic over K(u).
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  4. #4
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    let v be transcendental over K so that for some n>0.
    Need a bit more than that. What is important is that you can write your $\displaystyle f(x)$ under the form $\displaystyle q(x)+up(x)$ with $\displaystyle p(v)\neq 0$ and $\displaystyle q(x)\in K[x].$
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