To find the minimal polynomial and to show that (1-i) is an associate of (1+i) in given set

Question

I have two questions here:-

-Find the minimal polynomial of $$\frac{(1+\sqrt[3]{7})}{2}$$

-Show that $(1-i)$ is an associate of $(1+i)$ in $\mathbb{Q}(i)$

For the minimal polynomial I guess I should equate it to some variable x and by adjusting some terms and taking cube on both sides , the polynomial will be obtained.So please correct me here if am wrong

I don't have any idea to show how $(1-i)$ is associate of $(1+i)$ in $\mathbb{Q}(i)$.

Answer 1

For the first question, you are right. The answer will be $4x^3-6x^2+3x-4$.

And the minimal polynomial of $1+i$ is $x^2-2x+2$, of which $1-i$ is also a root.

Answer 2

For the second part, if you can't spot it straight away, $w$ and $z$ are associates if $w=uz$ where $u$ is a unit, hence $u=\frac wz$.

So take $\cfrac {1+i}{1-i}$, and see whether this is a unit.