TITLE: Why are there only 2 solutions for $x^n=1$?
QUESTION [6 upvotes]: (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?

REPLY [2 votes]: When you speek about solutions of an equation you must always be careful about the "algebraic structure" (In a specialistic language Fields/Rings) where you are searching a solution.
For example, searching a solution in $\mathbb{Z}$ is not equal to searching a solution over $\mathbb{Q}$ even in very simple cases.
An interesting example of this phenomenon is the polynomial $2x-1$.
The related equation has no solution in the ring $\mathbb{Z}$ but a unique solution in $\mathbb{Q}$ (that is in specialistic language $\mathbb{Z}$'s fraction field).
Looking more closely to your case, given a polynomial with just one variable (a so-called "univariate polynomial") of degree $n$, its associated equation has  always $n$ solution in (the field) $\mathbb{C}$ but it can have no solution in -(the field) $\mathbb{R}$
Look for example to the polynomial $x^2+1$.
Algebra in some sense is the study of this differences, but something sure are the following theorems:

Theorem Let $p(x)$ be a univariate polynomial of degree $n$. Then there always exist a "bigger" algebraic structure (called the splitting field of the polynomial p) where the equation $p(x)=0$ has exactely $n$ solutions (counted with multiplicity)

For example, in a case above, the field $\mathbb{C}$ is the splitting field of $x^2+1$ over $\mathbb{R}$
Well, to conclude, let's speack about $\mathbb{C}$.
It is a very "beautifull" algebraic structure because has a very nice property:

Theorem Let $p(x)$ an univariate polynomial of degree $n$ with coefficients in $\mathbb{C}$. Then $p(x)=0$ has exactely $n$ solution over $\mathbb{C}$ (counted with multiplicity)

In particular, since $\mathbb{R} \subset \mathbb{C}$, a real polynomial is a complex polinomial and its associated equation has always $n$ complex solution but cannot have $n$ real solutions.
This is precisely the case of $x^2+1$.
There are "not many" algebraic structure with the same property of $\mathbb{C}$.
This kind of structures are called "algebrically closed fields" and are one of the most important objects of Algebra.