Imaginary quadratic field Q(sqrt(-3))

The Imaginary quadratic field Q(sqrt(-3)) is the field extension

\mathbb Q(\sqrt{-3})=\mathbb Q[x]/(x^2+3)=\{a+b\sqrt{-3}:a,b\in\mathbb Q\},

with addition and multiplication induced by the displayed quadratic relation.

Scalar structure

Its ring of integers is the Eisenstein integer ring, generated by $(-1+\sqrt{-3})/2$ , rather than just by $\sqrt{-3}$ .

Maps and coordinates

A rational-linear endomorphism is any two-by-two rational matrix in the basis $1,\alpha$ , where $\alpha$ is the displayed quadratic generator. A unit-preserving rational-algebra map must send $\alpha$ to a root of its defining polynomial, so there are precisely the identity and conjugation. The invertible linear maps form $\mathrm{GL}_2(\mathbb Q)$ , a much larger group.

References

J. S. Milne, Algebraic Number Theory, Chapter 2, Example 2.41, p. 39: integral bases in quadratic fields.