Just a quick question here. I've noticed in my study of algebra that exponents are always based on the field of real numbers, and are typically integers.

If we have a field F of abstract elements is there an unambiguous and, hopefully, standard way of defining the expression $\displaystyle a^b$ where both a, b belong to F?

I have never seen this in any of the texts I've studied from but it would seem that, since many operations on the field of real numbers have been generalized to algebraic operations, the exponent operation would also have been generalized.

The only two exponents I can think of that would work would be to define 0 to be the additive identity and e the multiplicative identity of the field, then we have, for any element a (not equal to 0) of the field:

$\displaystyle a^0 = e$

and

$\displaystyle a^e = a$

and

$\displaystyle e^a = e$ <-- This last, of course, also works for a = 0

-Dan