Abstract Algebra: Theory and Applications · Problem 17.6.6

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · solving-cubic

Problem

Prove that the possible solutions for

z

in (4) are given by

\sqrt[3]{A}, \quad \omega\sqrt[3]{A}, \quad \omega^2\sqrt[3]{A}, \quad \sqrt[3]{B}, \quad \omega\sqrt[3]{B}, \quad \omega^2\sqrt[3]{B}

and use this result to show that the three possible solutions for

y

are

\omega^i \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{p^3}{27} + \frac{q^2}{4}}} + \omega^{2i} \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{p^3}{27} + \frac{q^2}{4}}},

where

i = 0, 1, 2