Abstract Algebra: Theory and Applications · Problem 18.4.1

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · units-in-gaussian-integers-variant

Problem

Let

z = a + b\sqrt{3}\,i

be in

\mathbb{Z}[\sqrt{3}\,i]

. If

a^2 + 3b^2 = 1

, show that

z

must be a unit. Show that the only units of

\mathbb{Z}[\sqrt{3}\,i]

are

1

and

-1

. Hint. Note that

z^{-1} = 1/(a+b\sqrt{3}\,i) = (a-b\sqrt{3}\,i)/(a^2+3b^2)

is in

\mathbb{Z}[\sqrt{3}\,i]

if and only if

a^2 + 3b^2 = 1

. The only integer solutions to the equation are

a = \pm 1, b = 0