Confused about exponent rule.

Question

I feel very dumb asking this. I'm trying to calculate $e^{\pi i n/4}$ for odd $n.$ I say the following: $e^{\pi i n/4} = (e^{\pi i n})^{1/4} = (-1)^{1/4}.$ However, Wolfram Alpha says that for $n = 5$ we have $-(-1)^{1/4}.$ I am confused.

Answer 1

The polynomial $p(x)=x^4+1$ splits into four linear factors over the algebraicely complete field $\mathbb{C}$:

$p(x)=(x-e^{i\frac{\pi}{4}})(x-e^{-i\frac{\pi}{4}})(x-e^{i\frac{5\pi}{4}})(x-e^{-i\frac{5\pi}{4}})$

and each of the corresponding roots could be taken as $``(-1)^{\frac{1}{4}}"$.

Clearly $e^{i\frac{5\pi}{4}}=-e^{i\frac{\pi}{4}}$ (This might explain why Wolfram found "the negative of Your root"). However since $\mathbb{C}$ is no ordered field there is no unique fourth root.

Answer 2

Let $n=2m+1$

$e^{i\pi (2m+1)/4}=e^{i\pi m/2}e^{i\pi /4}=\Big(\dfrac{1+i}{\sqrt 2}\Big)e^{\dfrac{i\pi m}{2}}$

Now put $m=2$