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Thread: Simple question on polynomial rings

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

    When we write $\displaystyle 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 $\displaystyle F[x_1, x_2, ... ... , x_n] $ is to check that the co-efficients belong to F and the indeterminates only contain $\displaystyle x_1, x_2, ... ... , x_n $.]

    OR

    when e write $\displaystyle 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 $\displaystyle 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 $\displaystyle F[x_1, x_2, ... ... , x_n] $ we can not reason about whether particular polynomials belong to $\displaystyle F[x_1, x_2, ... ... , x_n] $ until you know the exact nature of the ring $\displaystyle 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; Jun 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 $\displaystyle F[x_1,\ldots,x_n]$ literally means the ring of all polynomials in the indeterminates $\displaystyle x_1,\ldots,x_n$ with coefficients in $\displaystyle F$. 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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