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.
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.
Ok, so they are polynomials in. The proof that rational functions are a field will depend on the properties of the integral domain
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
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