# Quickest method of finding roots....

• May 18th 2009, 11:57 AM
AAM
Quickest method of finding roots....
Hi guys,

What is the quickest way to find the 4 complex roots of X^4+2=0 ?

I'm trying to find the splitting field for this equation over Q.

Many thanks in advance. :-) x
• May 18th 2009, 01:52 PM
HallsofIvy
Quote:

Originally Posted by AAM
Hi guys,

What is the quickest way to find the 4 complex roots of X^4+2=0 ?

I'm trying to find the splitting field for this equation over Q.

Many thanks in advance. :-) x

$x^4= -2$. Write -2 in polar form- $-2= 2 (cos(\pi)+ i sin(\pi))$ and use DeMoivre's formula.
• May 18th 2009, 10:04 PM
NonCommAlg
you can also write: $0=x^4+ 2 = (x^2 + \sqrt{2})^2 - 2\sqrt{2} x^2=(x^2 + \sqrt[4]{8} x + \sqrt{2})(x^2 - \sqrt[4]{8} x + \sqrt{2}).$