Answer to Question #179096 in Abstract Algebra for Anand

Question

Find all the units of Z[ √− 7].

Answer 1

Take x, y ∈ Z[ √− 7] such that x has the form of a+ b √− 7 while y takes the form c + d √− 7 for the arbitrary values of x and y.

if (a+ b √− 7) (c + d √− 7) = 1 then (a - b √− 7) (c - d √− 7) =1.

Therefore;

(a+ b √− 7) (c + d √− 7)(a - b √− 7) (c - d √− 7) = 1 since 1*1 = 1

thus;

(a² + 7b²)(c² + 7d²) = 1 hence

b² = d² = 0 and

a² = c² = 1 or -1 thus a=c = 1 or -1 which are the units in Z[ √− 7]