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How to prove the field extension is algebraically closed

  1. Apr 5, 2012 #1
    Suppose that E is a field extension of F, and every polynomial f(x) in F[x] has a root in E. Then E is algebraically closed, i.e. every polynomial f(x) in E[x] has a root in E.

    I've been told that this result is really difficult to prove, but it seems really intuitive so I find that surprising. Where can I find a proof of this result?

    Any help would be greatly appreciated.

    Thank You in Advance.
     
  2. jcsd
  3. Apr 5, 2012 #2


    No wonder you've been told that: it is false. Let [itex]E:= ℂ(t)[/itex] be the field of rational functions on the (transcendental) variable t. Then E is a field extension of [itex]ℂ[/itex] in which every pol. of [itex]ℂ[x][/itex] has a root, but E certainly is not alg. closed as, for example, the pol. [itex]x^2-t\in ℂ(t)[x][/itex] has no root in it...

    Now, the claim is true if one adds the condition that E/F is an algebraic extension.

    DonAntonio
     
  4. Apr 5, 2012 #3
    OK, so adding that condition where can I find the proof of the result?
     
  5. Apr 5, 2012 #4


    It's theorem 2 in http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/algclosure.pdf

    The proof isn't THAT easy but it isn't THAT hard, either.

    Enjoy, and for this read the whole paper. The first part is just the well-known and "standard" proof by Artin.

    DonAntonio
     
  6. Apr 5, 2012 #5

    mathwonk

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    my free algebra notes 844.I.1, do this in detail, in the first 12 pages.

    http://www.math.uga.edu/%7Eroy/844-1.pdf [Broken]
     
    Last edited by a moderator: May 5, 2017
  7. Apr 5, 2012 #6


    The question wasn't about the existence of alg. closures of fields, which Artin's proof, that also appears in your notes, does, but the fact that after the first construction of an extension E/F s.t. every non-constant pol. in E[x] has a root we can ALREADY stop there since F is alg. closed. I couldn't find this point in your notes, though perhaps I oversaw it.

    DonAntonio
     
    Last edited by a moderator: May 5, 2017
  8. Apr 7, 2012 #7

    mathwonk

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    oops, my mistake, thank you.
     
  9. Apr 8, 2012 #8
    Is it possible to prove the result without invoking the notion of perfect fields?
     
  10. Apr 8, 2012 #9


    I doubt it since the trick focuses on extensions K/F which can be put in the form [itex]K=F(\alpha)[/itex] , for some [itex]\alpha\in K[/itex] , and one

    achieves this using separability, but also: why? Separability shouldn't, imo, be a problem for someone trying to tackle what you're trying...

    DonAntonio
     
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