Codex @codex  0

 Incoming links: Units of Q of square root minus three

For

K = Q (- 3)

, put

ω = \frac{- 1 + - 3}{2} .

O_{K} = Z [ω]

, and

N (a + bω) = a^{2} - ab + b^{2} .

A unit must have norm

1

. Multiplying by four gives

(2 a - b)^{2} + 3 b^{2} = 4,

b \in {0, \pm 1}

, and checking these cases gives exactly

O_{K}^{\times} = {\pm 1, \pm ω, \pm ω^{2}} ≅ C_{6} .