Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 1995/07/10
Subject: Re: Cubic equations with real roots

rbury@delphi.com writes:

> The standard solution for cubic equations involves the cube root of
> imaginary numbers when there are three distinct real roots....
> Does anyone know of a formula for the
> solution of a cubic equation with real coefficients that works when
> there are three real roots and involving only the basic arithmetic
> operations and the extraction of roots of real numbers?

No, no one does, because there is no such formula.  In particular, if
a cubic equation with rational coefficients has three real irrational
roots, then the roots are not expressible in real radicals.

I unfortunately don't know the proof of this, nor can I give you an
exact reference.  But the topic is called "equation theory", and there
were a lot of books written about it about 70-100 years ago.  I've seen
this stated, with a proof that I didn't understand, in such a book.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil