can any one give a intuitive elaboration for "a field of rational fractions"? per wiki a field is 

a set on which addition, subtraction, multiplication, and division are
  defined, and behave as the corresponding operations on rational and
  real numbers do.

this section gives some more concrete examples ${\displaystyle {\frac {b}{a}}\cdot {\frac {a}{b}}={\frac {ba}{ab}}=1.}$
In the context of a group of rational numbers $\dfrac{a}{b}$, $\dfrac{b}{a}$, b ≠ 0, a ≠ 0
does a field of this group of rational numbers have and only have following operations
addition: ${\displaystyle{\frac {b}{a}} + {\frac {a}{b}}}$
subtraction: ${\displaystyle{\frac {b}{a}} - {\frac {a}{b}}}$, ${\displaystyle{{\frac {a}{b}}} - \frac {b}{a}}$
multiplication: ${\displaystyle{\frac {b}{a}}\cdot {\frac {a}{b}}}$
division: $\dfrac{b/a}{a/b}$, $\dfrac{a/b}{b/a}$
 A: A field is a context in which you can do arithmetic and in which linear equations have solutions - you can divide, except by zero. 
Ordinary fractions made out of integers form the field of rational numbers $\mathbb Q$, and the numbers which make up the "number line" are the field of real numbers $\mathbb R$.
I think that the most confusing word in your question is "fractions" - some fields like $\mathbb Q$ arise from constructing fractions - in this case, fractions from the integers $\mathbb Z$. Here we are constructing a field from a "ring" in which addition, subtraction and multiplication are possible, but division is not - if we try to divide $3$ by $2$ we don't get an integer result, but if we allow fractions, everything works.
So introducing the idea of fractions means we have another structure hiding in the background from which the fractions are built - and I think one issue for your question is that this hidden structure is not defined. Defining fractions also has a subtlety in that care must be taken not ever to divide by zero, and making sure that doesn't happen (in a general context) does require accurate definition of equality of fractions.
Another flaw in the approach to defining via fractions is that not every field naturally arises in this way. The real numbers, for example, are not defined as fractions. They are an extension of the rationals (the real numbers contain the rational numbers), and extensions of fields are important. Then the finite field of integers modulo $5$ (we could replace $5$ by any other prime number) is not naturally expressed as a field of fractions. The construction "modulo $5$" in fact works in any "ring" if we work "modulo a maximal ideal" - and that brings in extra ideas too.
Fractions are just one example of a field constructed from a ring - just one example of taking an algebraic structure in which division is not possible, and modifying it so that division becomes possible. Once we can divide we can solve linear equations.
Sometimes there is a two stage process - if we extend a field by adding a symbol $x$ and creating polynomials, we can define multiplication and addition and subtraction easily enough, and we get a "ring" of polynomials. Then we can take fractions to obtain a field of rational functions. Or we can factor out modulo a maximal ideal and get an "algebraic extension". Algebraic extensions are important in studying the solutions to polynomials.
So, in summary, a field is a rich algebraic idea which extends the methods of ordinary arithmetic to other contexts. Not every field arises as "fractions" and the ideas of a "ring", "modulo an ideal" and extensions, including polynomial extensions and algebraic extensions are also important.
