Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as $x=f(-2)$, but I don't know how to find it "by hand". This led me to think about $\sqrt{2}$ roots of unity (since the equation is of the form $z^{\sqrt{2}}=1$, and in general irrational roots of unity. I want to know if they exist.
Thanks!