Is a square root of a unit of a quadratic field also a unit?

Question

For example: $\,$In $\,$ $\mathbb{Z}[(1+\sqrt{5})/2]$, $\;\omega^3-\omega-1$ $\,$ is a unit. For: $\,$$\omega=\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}$ ; his Norm is 1. Then:$\,$Is $\,$$\,$ $\sqrt{\omega^3-\omega-1}$ $\,$ also a unit of$\,$ $\mathbb{Z}[(1+\sqrt{5})/2]$$\,$?$\,$ Thanks in advance

Answer 1

You should expect that the square root of a unit will not be in the given field.

But if your unit $\zeta$ is root of the monic $\Bbb Z$-polynomial $f(X)$, then you see: first, that $\zeta^{1/2}$ is root of $f(X^2)$; second, that $\zeta^{1/2}$ is root of a monic $\Bbb Z$-polynomial, thus an algebraic integer; third, that since the constant coefficient of $f$ is $\pm1$, $\zeta^{1/2}$ is also a unit.