Algebraic Number/Examples/Cube Root of 4 minus 2 i

Example of Algebraic Number

$\sqrt [3] 4 - 2 i$ is an algebraic number.

Let $x = \sqrt [3] 4 - 2 i$.

We have that:

$\ds x + 2 i$

$=$

$\ds \sqrt [3] 4$

$\ds \leadsto \ \ $

$\ds \paren {x + 2 i}^3$

$=$

$\ds 4$

$\ds \leadsto \ \ $

$\ds x^3 + 6 i x^2 + 12 i^2 x + 8 i^3$

$=$

$\ds 3$

$\ds \leadsto \ \ $

$\ds x^3 - 12 x - 4$

$=$

$\ds i \paren {8 - 6 x^2}$

$\ds \leadsto \ \ $

$\ds \paren {x^3 - 12 x - 4}^2$

$=$

$\ds -\paren {8 - 6 x^2}^2$

$\ds \leadsto \ \ $

$\ds x^6 - 24 x^4 - 8 x^3 + 144 x^2 + 96 x + 16$

$=$

$\ds -\paren {64 - 96 x^2 + 36 x^4}$

$\ds \leadsto \ \ $

$\ds x^6 + 12 x^4 - 8 x^3 + 48 x^2 + 96 x + 80$

$=$

$\ds 0$

Thus $\sqrt [3] 4 - 2 i$ is a root of $x^6 + 12 x^4 - 8 x^3 + 48 x^2 + 96 x + 80 = 0$.

Hence the result by definition of algebraic number.

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $47 \ \text {(b)}$