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Math Help - Variable of ring of polynomials

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    Variable of ring of polynomials

    Hi ... I just needed clarification.

    Say we have a ring R, and we form the polynomial ring R[x]. What are the requirements/restrictions for the variable x?

    I've been doing some reading, and from what I'm getting, it can essentially be anything ... is this right?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bingk View Post
    Hi ... I just needed clarification.

    Say we have a ring R, and we form the polynomial ring R[x]. What are the requirements/restrictions for the variable x?

    I've been doing some reading, and from what I'm getting, it can essentially be anything ... is this right?
    What do you mean by restriction of variable?
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    For example, if R is the ring of integers, then does x have to be integers also? or can it be complex numbers as well?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bingk View Post
    For example, if R is the ring of integers, then does x have to be integers also? or can it be complex numbers as well?
    I think you're misunderstanding the notation. You don't actually put values in for x, they're just "place-holders" in a sense.
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    Quote Originally Posted by Bingk View Post
    Hi ... I just needed clarification.

    Say we have a ring R, and we form the polynomial ring R[x]. What are the requirements/restrictions for the variable x?

    I've been doing some reading, and from what I'm getting, it can essentially be anything ... is this right?

    You can think of x as merely a symbol or, if you want, what's also called sometimes a dummy variable. In fact it must be an element transcendental over R, meaning: no non-zero polynomial with coefficients in R vanishes at x.

    Tonio
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    I get that it's a variable, but I was wondering what the "domain" is ... which you kinda answered. I did some reading, I get what a transcendental element is ... so just to be clear, x should be transcendental over R?

    If x is not transcendental over R, what would happen?

    I'm sorry if these are basic questions, but I'm learning this on my own, no classes or qualified teachers to help me out
    Last edited by Bingk; February 24th 2010 at 06:43 AM. Reason: Previous post obsolete
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