I'm stuck on the following exercise:
Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such that $a=b^p$, or $4$ divides $n$ and $a=-4b^4$.
One direction is not difficult to show, as either $X^{\tfrac{n}{p}}-b$ is a factor of $X^n-a$, or we have $$(X^n-a)=(X^{\tfrac{n}{2}}+2bX^{\tfrac{n}{4}}+2b^2)(X^{\tfrac{n}{2}}-2bX^{\tfrac{n}{4}}+2b^2),$$ for the two given cases. However, I'm having trouble proving the other direction. Any hints as to how to go about this are very welcome.