If is a field in which , prove that is not a rational function. Hint. Mimic the classical proof that is irrational
It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!
Suppose that is a rational function in . Then there exist polynomials and that are relatively prime with ; clearly we may take . Then or , which says , and since , we can infer . Moreover, since and , then or , which contradicts the fact that the polynomials are relatively prime. Hence cannot be a rational function.
How does this sound? Aside from the problem of discussing – in the context of an arbitrary field, I am worried about not using the fact that , at least not explicitly. Where exactly is this assumed used, if at all?
Suppose that , and let . Then . If , and therefore must be a unit. Letting we get ; and letting we get . Since we are working in a field, there can be no nonzero nilpotent which means that and are both zero. Adding the two equations together yields , and since , which contradicts the fact that is a unit.
How does this sound?
If , then implies . Yet if , we get and therefore since there are no nonzero nilpotent elements in a field; similarly, . Since , has two distinct roots in yet is only a 1-st degree polynomial, a contradiction.