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Tuesday, April 29, 2008

The field of fractions of an integral domain

V=\mathbb{Z} \times \mathbb{Z} \backslash \left\{0\right\} \\<br />R: (a,b)\equiv (c,d) \Leftrightarrow  ad=bc\\<br />\\<br />(a,b) + (c,d) = (ad + bc, bd)\\<br />(a,b) \cdot (c,d) = (ac, bd)

Create a set V of ordered pairs from {..., -2, -1, 0, 1, 2, ...} (integers) and {..., -2, -1, 1, 2, ...} (integers excluding 0). Elements of V are for example (3,1), (5,1) and (4,2).

Create an equivalence relation on elements of V. Two elements (a,b) and (c,d) are 'equivalent', 'belong to the same equivalence class' if ad=bc. For example (4,2) and (8,4) are equivalent while (4,1) and (8,4) are not.

Define addition '+' as (a,b) + (c,d) = (ad + bc, bd).

Define multiplication '.' as (a,b) . (c,d) = (ac, bd).

This is how the field of Rationals is formally constructed from the Integers.

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