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Math Help - Simple question on polynomial rings

  1. #1
    Super Member Bernhard's Avatar
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    Simple question on polynomial rings

    When we write  F[x_1, x_2, ... ... , x_n] where F is, say, a field, do we necessarily mean the set of all possible polynomials in x_1, x_2, ... ... x_n with coefficients in F? [In this case, essentially all that is required to determine whether a polynomial belongs to  F[x_1, x_2, ... ... , x_n] is to check that the co-efficients belong to F and the indeterminates only contain  x_1, x_2, ... ... , x_n .]

    OR

    when e write  F[x_1, x_2, ... ... , x_n] do we mean to include possible cases such as the set of polynomials with even coefficients - that is we may be talking about the set of polynomials with even co-efficients - so we cannot be sure what ring of polynomials we are talking about when we write  F[x_1, x_2, ... ... , x_n] until we specify the exact nature of ring of polynomials we are talking about further.


    If the latter is the case when given  F[x_1, x_2, ... ... , x_n] we can not reason about whether particular polynomials belong to  F[x_1, x_2, ... ... , x_n] until you know the exact nature of the ring  F[x_1, x_2, ... ... , x_n]

    I very much suspect that the former is the case but ... ... Can someone please confirm or clarify this.

    Peter
    Last edited by Bernhard; June 30th 2013 at 02:56 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Simple question on polynomial rings

    Hello,

    I am a little confused as to how you would have this confusion. You are correct about the former--the ring F[x_1,\ldots,x_n] literally means the ring of all polynomials in the indeterminates x_1,\ldots,x_n with coefficients in F. Things in math are very rarely wishy-washy enough so that this symbol could be as ambiguous as you claim.

    I hope this helps.

    PS Evenness makes no sense in the context of fields.
    Thanks from Bernhard
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