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Math Help - subfield isomorphism

  1. #1
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    subfield isomorphism

    How do I show that if p > 0 is the characteristic of the finite field F (p is prime obviously) that F contains a subfield {k * 1F: k=0,1,...,p-1} that is isomorphic to Zp?

    Note k*1F= 1+1+1...+1 (k times)
    Thank you in advance.

    Also, how can I show that the set {k * 1F: k=0,1,...,p-1} actually is a subfield?
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    How do I show that if p > 0 is the characteristic of the finite field F (p is prime obviously) that F contains a subfield {k * 1F: k=0,1,...,p-1} that is isomorphic to Zp?
    This is one of the elegant results about fields.

    Theorem: If R is a ring with unity and with charachteristic n>0, then R contains \mathbb{Z}_n as a subring up to isomorphism.

    Thus, given a field with a charachetristic n>1 then n must be a prime. Because otherwise it will have zero-divisors, which a field does not have. Thus it contains \mathbb{Z}_p as a subfield up to isomorphism.

    Quote Originally Posted by PvtBillPilgrim View Post
    Also, how can I show that the set {k * 1F: k=0,1,...,p-1} actually is a subfield?
    This is a known fact that \mathbb{Z}_p is always a field. Because it is a finite integral domain, and a finite integral domain is always a field.
    Last edited by ThePerfectHacker; January 28th 2007 at 05:07 AM.
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  3. #3
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    How do you prove that initial part though (the theorem)?
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    How do you prove that initial part though (the theorem)?
    The map \phi:\mathbb{Z}\to R which we define as \phi(m)=m\cdot 1 is a homomorphism (excerise, prove it). By the fundamental theorem the kernel is an ideal of \mathbb{Z}. Now all ideals of \mathbb{Z} are of the form k\mathbb{Z} because a subgroup of a cyclic group is cyclic. Thus, if R has charachteristic n>0 then, \ker \phi =n\mathbb{Z}. Again by the fundamental theorem, \phi [\mathbb{Z} ] \simeq \mathbb{Z}/n\mathbb{Z}\simeq \mathbb{Z}_n. Thus, the subring \phi[\mathbb{Z}] confirms the theorem. Q.E.D.

    What is happening to you in this class? I seriously doubt it they will ask you to prove this theorem, is it not in your textbook?
    Last edited by ThePerfectHacker; January 27th 2007 at 08:06 PM.
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    Yeah, we have no textbook.

    Thanks for the help.
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  6. #6
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    Quote Originally Posted by PvtBillPilgrim View Post
    Yeah, we have no textbook.
    Why not? Is the professor insane?
    Last edited by ThePerfectHacker; January 28th 2007 at 05:18 AM.
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