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.
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 $\displaystyle \mathbb{R}[x]$. The proof that rational functions are a field will depend on the properties of the integral domain $\displaystyle \mathbb{R}[x]$.
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 $\displaystyle \mathbb{R}[x]$) 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.