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Math Help - Prime Ideals of Extension Fields

  1. #1
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    Prime Ideals of Extension Fields

    Let E/F be an extension of fields. For a in E let phi_a be the evaluation homomorphism defined by: phi_a: F[x] --> E; f --> f(a)
    Show that for all a in E, ker(phi_a) is a prime ideal in F[x].
    I know that ker(phi_a) is an ideal of F[x], I'm having trouble showing that it is prime.
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    Let E/F be an extension of fields. For a in E let phi_a be the evaluation homomorphism defined by: phi_a: F[x] --> E; f --> f(a)
    Show that for all a in E, ker(phi_a) is a prime ideal in F[x].
    I know that ker(phi_a) is an ideal of F[x], I'm having trouble showing that it is prime.
    \frac{F[x]}{\ker \phi} is isomorphic to a subring of E and hence it has to be an integral domain because E is a field and thus an integral domain. so \ker \phi is a prime ideal of F[x].
    Last edited by NonCommAlg; April 14th 2009 at 07:59 PM.
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