3.I.2 F3 . \mathrm{I} . 2 \mathrm{~F} \quad

Groups, Rings and Modules | Part IB, 2004

Let RR be the subring of all zz in C\mathbb{C} of the form

z=a+b32z=\frac{a+b \sqrt{-3}}{2}

where aa and bb are in Z\mathbb{Z} and ab(mod2)a \equiv b(\bmod 2). Prove that N(z)=zzˉN(z)=z \bar{z} is a non-negative element of Z\mathbb{Z}, for all zz in RR. Prove that the multiplicative group of units of RR has order 6 . Prove that 7R7 R is the intersection of two prime ideals of RR.

[You may assume that RR is a unique factorization domain.]

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