Results 1 to 4 of 4

Math Help - Field Extensions: Algebraic and Transcendental Elements

  1. #1
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485

    Field Extensions: Algebraic and Transcendental Elements

    Hello there. I have the following problem:

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

    I've tried writing down the minimal polynomial for v over 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    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:

    \mu_{P,K(u)}\in K(u)[X]-K[X] . I think it is sufficient to conclude
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2009
    Posts
    192
    Thanks
    4
    Not sure if this is right but have a look at it see what you think;

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

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

    then
    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 a_{i_j},v^{i_j} \in K(v) so a_{i_j}v^{i_j} \in K(v)

    so letting

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

    where

    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).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    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 f(x) under the form q(x)+up(x) with p(v)\neq 0 and q(x)\in K[x].
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Algebraic Extensions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 6th 2011, 01:10 AM
  2. Replies: 7
    Last Post: April 14th 2010, 07:09 AM
  3. Field Extensions
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 6th 2009, 12:34 AM
  4. Field Extensions
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 3rd 2008, 03:30 PM
  5. Field Extensions
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: March 18th 2006, 09:27 AM

Search Tags


/mathhelpforum @mathhelpforum