1. Exponents

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 $a^b$ 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:
$a^0 = e$
and
$a^e = a$
and
$e^a = e$ <-- This last, of course, also works for a = 0

-Dan

2. I cannot imagine how that is possible.

Given an algebraically closed field $K$ how does one even define $a^{1/2}$ or $a^{1/3}$?

I seen a book on Number Theory write a fractional exponent when working in an arbitrary field, but I assumed all it meant is that it did not matter what root you use.

So the answer to your question is probably not. I never seen anything like that happen.

3. Originally Posted by ThePerfectHacker
I cannot imagine how that is possible.

Given an algebraically closed field $K$ how does one even define $a^{1/2}$ or $a^{1/3}$?

I seen a book on Number Theory write a fractional exponent when working in an arbitrary field, but I assumed all it meant is that it did not matter what root you use.

So the answer to your question is probably not. I never seen anything like that happen.
Hmmmm.... That's a pity. Perhaps this is how I will become famous...

Nah!

-Dan

4. Originally Posted by topsquark
Hmmmm.... That's a pity. Perhaps this is how I will become famous...

Nah!

-Dan
The farthest I can go are complex exponents. Which I find really fun, because many people get so confused with the mutiple of $2\pi i$ thing.

5. Hm... Just for the reals, to define exponents requires the topological principle of completeness and a limiting process. No algebraist I know likes either of these concepts! They are of purely analytical nature.

I don't doubt there are cases where exponents can work. But the general case would have to use tools beyond the average Algebraic ones.

6. Originally Posted by Rebesques
Hm... Just for the reals, to define exponents requires the topological principle of completeness and a limiting process. No algebraist I know likes either of these concepts! They are of purely analytical nature.
What is so hard?

$a^b = \exp (b\ln a)$ where $a>0$? Unless your point is, that what I just did was purely analytic. I agree.

7. that what I just did was purely analytic. I agree

That makes two of us Just to define powers of e, one need all of the forementioned analytic tools.