representation of rational field I want to  know how is represented general form of rational field, for example definition of
${\mathbb Q}(\sqrt{2})$ is represented as $p+q \sqrt{2}$, where $p$ and  $q$ are rational numbers, for example let us consider following case
${\mathbb Q}(\sqrt{2},\sqrt{3})$ is represented by  
$$q_1 + q_2\sqrt{2} + q_3\sqrt{3} + q_4\sqrt{6}$$
My question is  in general how it is represented  
${\mathbb Q}(q_1,q_2,q_3,...q_n)$  where  $q_1$ can be  rational or irrational number, thanks in advance
 A: The answer to your question is much more complicated than you might think, which is either going to be bad news for you ( if you wanted to understand how field extensions work in five minutes ), or really good news for you ( if you were looking for a seriously rich and interesting avenue of study ).  
Very very brief version:
1) if any of the $q_i$ are rational, you can ignore them.  They don't matter. if $q_1$ is rational, then $\mathbb{Q}(q_1,q_2)=\mathbb{Q}(q_2)$
2) The next important distinction is whether or not the $q_i$ are algebraic, ie., if they each satisfy a ( potentially different ) polynomial which coefficients in $\mathbb{Q}$.  If they all do, then you can say that every element of your field can be written as a linear combination of products of the generators.  Even more, if $q_i$ satisfies a polynomial of degree $n_i$, then the elements of your field can be written as linear combinations of $q_1^{m_1}\cdot\ldots\cdot q_r^{m_r}$ with $m_i \leq n_i$.  This is the case for $\mathbb{Q}( \sqrt{2}, \sqrt{3} )$ in your example.  The expressions are unlikely to be unique, though (although they are in the case of $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ ).  
3) If some of the $q_i$ are transcendental.... well, then you're pretty much stuck with Jyrki's comment that the elements of the field are rational functions of the generators.  
