Algebraic Integers in Cyclotomic fields

Question

Let $\zeta$ be a primitive 8th root of unity in $\mathbb{C}$ and let $\alpha = \frac{1-\zeta}{2}$

(i) Determine the minimal polynomial of $\alpha$ over $\mathbb{Q}$

(ii) Is $\alpha^{-1}$ an algebraic integer?

I think I have part (i) sorted since I know $\zeta^{8} = 1$ so I have the minimal polynomial of $\zeta = x^4 +1$. So the minimal polynomial of $\alpha$ over $\mathbb{Q}$ should be $(1-2x)^4 +1$ ? Would love confirmation on that though!

I'm not sure about part (ii) though... Any help you can provide would be greatly appreciated!

Answer 1

For part (i), your work looks good to me! Beware, some people require minimal polynomials to be monic, in which case you can just expand out the expression you have and divide by the leading coefficient.

For (ii), I'll give you a hint first: what's the norm of $1 - \zeta$ for the extension $\mathbb{Q}(\zeta)/\mathbb{Q}$?

Note that the minimal polynomial of $\zeta$ is $$m(x) = x^4 +1 = (x-\zeta)(x-\zeta^3)(x-\zeta^5)(x-\zeta^7).$$ Thus, we have $$ \mathrm{Nm}(1-\zeta) = (1-\zeta)(1-\zeta^3)(1-\zeta^5)(1-\zeta^7) = m(1) = 2, $$ so that $$ \frac{1-\zeta}{2} \cdot (1-\zeta^3)(1-\zeta^5)(1-\zeta^7) = 1. $$