# Find all units in the ring $\Bbb Z[\omega]$ where $\omega$ is the primitive $p^{th}$ root of unity.

Let $$p>0$$ be a prime and let $$\omega$$ be a primitive $$p^{\text{th}}$$ root of unity. I am trying to find all units in the ring $$\Bbb Z[\omega]$$.

Every element of $$\Bbb Z[\omega]$$ is of the form $$z=a_1\omega+a_2\omega^2+\dots+a_{p-1}\omega^{p-1}$$ for some $$a_1,\dots,a_{p-1}\in\Bbb Z$$. Attempting to find the form of the inverse $$z$$ one could try to make the denominator real by multiplying the complex conjugate. Note that $$\overline{\omega}=\omega^{p-1}$$ so that $$\begin{multline} \frac{1}{a_1\omega + a_2\omega^2 + \dots + a_{p-1}\omega^{p-1}} \\[4pt] = \frac{a_1\omega^{p-1} + a_2\omega^{2(p-1)} + \dots + a_{p-1}\omega^{(p-1)(p-1)}} {(a_1\omega + a_2\omega^2 + \dots + a_{p-1}\omega^{p-1})(a_1\omega^{p-1} + a_2\omega^{2(p-1)} + \dots + a_{p-1}\omega^{(p-1)(p-1)})} \end{multline}$$ but note that $$\omega^{k(p-1)} = \omega^{p-k}$$ so the later fraction is $$\frac{a_1\omega^{p-1} + a_2\omega^{p-2} + \dots + a_{p-1}\omega} {(a_1\omega + a_2\omega^2 + \dots + a_{p-1}\omega^{p-1}) (a_1\omega^{p-1} + a_2\omega^{p-2} + \dots + a_{p-1}\omega)}.$$

I have trouble simplifying the denominator further, so I took the example where $$p=3$$ where $$\omega$$ is then a primitive $$3^{\text{rd}}$$ root of unity. If $$z=a+b\omega$$, the fraction above then is easy to calculate as $$\frac{a + b\omega}{a^2 - ab + b^2}.$$ Since $$\{1,\omega\}$$ is linearly independent over $$\Bbb Q$$, $$a^2-ab+b^2$$ must divide $$\gcd(a,b)$$ but $$\gcd(a,b) = \pm 1$$ or else $$a+b\omega$$ cannot be a unit. We conclude that $$a^2-ab+b^2 = \pm 1$$ and an elementary argument shows that the only possibilities are $$(a,b)=(0,1)$$, $$(0,-1)$$, $$(1,0)$$, $$(-1,0)$$, $$(1,1)$$, or $$(-1, -1)$$.

How can I reach a similar conclusion in the general case of the prime $$p$$?

• There will be lots more units as $p$ grows—roughly a $\frac{p-3}2$-parameter family of units, according to Dirichlet's unit theorem. An important class of such units are the cyclotomic units. Oct 18, 2023 at 21:41