Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 2F (Groups, Rings and Modules)

Write $\tau =(1+\sqrt{-3})/2$. The parity condition gives $R=\mathbb{Z}[\tau ]$, since $(a+b\sqrt{-3})/2=(a-b)/2+b\tau$. For any such element,If $a,b$ are even the numerator is divisible by four; if they are odd it is congruent to $1+3=0$ modulo four. Therefore $N(z)$ is a nonnegative integer. It is zero only at zero, and the Eisenstein-integer norm is multiplicative because complex conjugation respects multiplication.

If $z$ is a unit, $N(z)N(z^{-1})=1$, so $N(z)=1$. Conversely $N(z)=1$ makes $\overline{z}$ its inverse in $R$. The equation $a^2+3b^2=4$ permits exactly $(a,b)=(\pm 2,0)$ or $(\pm 1,\pm 1)$. HenceIndeed ${\tau }^2=\tau -1$ and ${\tau }^3=-1$, so this unit group is cyclic of order six.

For the prime-ideal assertion, ${\tau }^2-\tau +1=0$ and the polynomial $t^2-t+1$ has distinct roots $3,5$ modulo seven. Consequently evaluation at either root defines a surjective ring map $R\to {\mathbb{F}}_7$. Its kernelsare distinct maximal ideals, hence prime ideals. If $A+B\tau$ belongs to both, then $A+3B\equiv A+5B\equiv 0(\mathrm{mod}7)$. Subtraction gives $2B\equiv 0$, so both $A$ and $B$ are divisible by seven. The converse inclusion is immediate, provingEquivalently, the two distinct factors of the reduced polynomial give the Chinese remainder theorem decomposition $R/7R\cong {\mathbb{F}}_7\times {\mathbb{F}}_7$. This proof does not require the permitted unique-factorization assumption.