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I was studying gaussian integers recently and I started wondering if exponentiation with complex exponent could be meaningful defined in modular arithmetic.

What I mean is, if one defines

$$ i^2 = -1 \mod(n) $$

can then be also defined what

$$ a^i \mod(n) $$ is?

I looked around and I could not find any definitions or rules of how this should work (if it works at all). I also tried by setting $a^i=R(a)+I(a)i \mod(n)$ with $R(\cdot)$ and $I(\cdot)$ being some functions that return integer, to see if I can come up with any idea what these functions could be. Now, all I have is A4 sheet of identities, which gave me no real insight what those functions could be.

Since we are in modulrat arithmetic, I cannot use trigonometric functions or euler identity like in normal complex number, but I also could not find if there exist any similar expression for gaussian integers.

So, my question is if anyone knows if there exist a theory that deals with this problem? Even if it is not totaly general but it works only for certian $n$s (e.g. primes) that's fine with me.

Klemen Zajc
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    When computing powers of $,a\pmod{! n},,$ exponent arithmetic is well defined only modulo the order of $,a,$ (or some multiple of it), assuming $,a,$ is coprime to $,n.,$ For example, if $,a,$ has order $,5,$ then $!\bmod 5!:,\ 2^2\equiv -1,$ so $,i := 2,$ is a square root of $,-1,,$ hence $,(a^i)^i \equiv a^{-1}\pmod n,,$ and $$\large (a+bi)(c+di)\equiv e+fi !!!\pmod{! 5} \ \Rightarrow\ (a^{a+bi})^{c+di}\equiv a^{e+fi}!!!\pmod{!n}\qquad\qquad$$ – Bill Dubuque Dec 20 '21 at 12:22
  • See here for another example of using square-roots (quadratic algebraic integers) in modular arithmetic. These matters will be clarified if you master the universal property of quotient rings. – Bill Dubuque Dec 20 '21 at 13:13
  • @BillDubuque, thanks that was helpful already. In your example you could actually calculated $i:=2$. But what if we have an order, where $-1$ is a nonresidue? I was thinking of this case since it is somewhat analogous to $i^2=-1$ in normal arithmetic (where $i$ also does not have real value, yet we can use it in calculations). Could this work and does it have any sense? – Klemen Zajc Dec 20 '21 at 14:51

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