Quickest method of finding roots....

**AAM** · May 18th 2009, 10:57 AM

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

**HallsofIvy** · May 18th 2009, 12:52 PM

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.

**NonCommAlg** · May 18th 2009, 09:04 PM

you can also write: $0=x^4+ 2 = (x^2 + \sqrt{2})^2 - 2\sqrt{2} x^2=(x^2 + \sqrt[4]{8} x + \sqrt{2})(x^2 - \sqrt[4]{8} x + \sqrt{2}).$

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