Notes for Elements of Abstract Algebra

Introduction

$x^3+qx-r=0$

Let $x=u+v⟹(u^3+3u^{2}v+3u{v}^2+v^3)+q(u+v)-r=0$ $⟹u^3+v^3+(3uv+q)(u+v)-r=0$

Choose $u,v$ so that $3uv+q=0⟹uv=-{\displaystyle \frac{q}{3}}⟹v=-{\displaystyle \frac{q}{3u}}$

$⟹u^6-r{u}^3-{\displaystyle \frac{q^3}{27}}=0⟹u^3={\displaystyle \frac{r}{2}}\pm \sqrt{{\displaystyle \frac{r^2}{4}}+{\displaystyle \frac{q^3}{27}}}$

Let $\omega =e^{i\frac{\pi }{3}}=-{\displaystyle \frac{1}{2}}+i{\displaystyle \frac{\sqrt{3}}{2}},{\omega }^2=e^{-i\frac{\pi }{3}}=-{\displaystyle \frac{1}{2}}-i{\displaystyle \frac{\sqrt{3}}{2}}$, then three roots of original equation are:

$u+v,\omega u+{\omega }^{2}v,{\omega }^{2}u+\omega v$

Chapter 1 - Set Theory