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Math Help - field question help please

  1. #1
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    field question help please

    R(x)={P(x)/Q(x): p and q are polynomials} can you explain me how can i show this is a field?i reguest explain me step bye step please.thanks for your helps.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Your statement is very vague. These are polynomials over what? An integral domain? An arbitrary ring? A field? The real numbers? It's very important that you specify.

    Do you know the axioms for a field? If so, and you know what your polynomials are, you just have to verify each of the axioms : show that it is an abelian group under addition, that the nonzero elements form an abelian group under multiplication, that multiplication is distributive over addition, etc.
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  3. #3
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    these are polynomials as in given set {sum i=1-->n(ai.xi)| ai element of R}
    i am asking how to show this is a field you are asking is this an integral domain,don't i have to show that this is an integral domain?and in my book there is a given theorem says:every field is an integral domain.i think i am confused when i translate in english.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Ok, so they are polynomials in \mathbb{R}[x]. The proof that rational functions are a field will depend on the properties of the integral domain \mathbb{R}[x].

    Take a look at the various defining properties of a field here. Verify them one by one for the set of rational functions (quotient of polynomials in \mathbb{R}[x]) and your problem is solved; none of them is hard to establish. If you have trouble with one of them feel free to ask again.
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