# Math Help - factorization of x^5-4

1. ## factorization of x^5-4

Hey guys.

I have an exam tomorrow, and one of my practice problems asks me to find the splitting field (and its degree) of $x^5-4$ over $\mathbb{Q}$.

Clearly $u=4^{1/5}$ is a root, and

$x^5-4=(x-u)(x^4+ux^3+u^2x^2+u^3x+u^4)$.

Beyond that, though, I don't know where to go. I might just be missing something deceptively simple. Anyway, some assistance would be much appreciated!

2. x^5=4 over complex field C contains 1 real root and 4 complex roots.

3. Originally Posted by Also sprach Zarathustra
x^5=4 over complex field C contains 1 real root and 4 complex roots.
Thank you, but what are those roots? That's what's giving me trouble.

4. Originally Posted by hatsoff
Thank you, but what are those roots? That's what's giving me trouble.

The roots are $4^{1/5}, w 4^{1/5}, w^24^{1/5}, w^34^{1/5}, w^44^{1/5}$, with $w=$ a primitive root of 1 of order 5.

Tonio