Is it possible to construct a field over the irrationals?
The problem is to define two binary operations that satisfy the axioms of a field.
The regular multiplication and addition operators fail to satisfy these axioms. For example:
Multiplication: $$\sqrt{2}\cdot \sqrt{2}=2\notin\mathbb{R}\setminus\mathbb{Q}$$
Addion: $$\sqrt{2}-\sqrt{2}=0\notin\mathbb{R}\setminus\mathbb{Q} $$
Do you have an example?