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
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.