TITLE: How to find the roots of $x^4 +1$
QUESTION [12 upvotes]: I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ is a positive term as the equation above.
Thanks

REPLY [4 votes]: Because $x^8-1=(x^4+1)(x^4-1)$, the roots of $x^4+1$ are the roots of $x^8-1$ that are not roots of $x^4-1$.
The roots of $x^n-1$ are easy: $e^{\frac{2k\pi i}{n}}$ for $k=0,\dots,n-1$.
Therefore, roots of $x^4+1$ are $e^{\frac{2k\pi i}{8}}$ for $k$ odd.