Answer to Question #183510 in Abstract Algebra for Anand

Question

Find all the units of Z[ √-7].

Answer 1

Let x,y $\in$ Z[-7]

Then x has the form a+b$\sqrt{-7}$ and y has the form c+d$\sqrt{-7}$ for arbitrary x and y.

$If\\ (a+b\sqrt{-7})(c+d\sqrt{-7})=1 , then\\ (a-b\sqrt{-7})(c-d\sqrt{-7})=1, so 1=(a+b\sqrt{-7})(c+d\sqrt{-7})(a-b\sqrt{-7})(c-d\sqrt{-7})=1\\ \text{which gives}\\ 1=(a^2+7b^2)(c^2+7d^2)\\ So,\\ b=d=0,and \\ 1=a^2c^2,\\ \text{so either}\\ a=c=1, or, a=c=-1\\ \text{and these are the only units in}Z\sqrt{-7}$