Why are there only 2 solutions for $x^n=1$? (where $n>0$)
I have been taught that an equation with the highest power $n$ will always have $n$ solutions. This does not appear to be the case with:
$$x^n=1 \implies x=\pm1$$
Where $n$ is even, the solutions are $1$ and $-1$; where $n$ is odd, the only solution is $1$.
Why? Is it a special case, or is there something I'm missing?

It has $n$ roots, but only if you consider roots $x_i\in \Bbb C$. These "special" ones are calleds the roots of unity.
As an example, consider $X^4=1$; try to check that $X_1=i$ is a root of said equation.
