What is the general form of elements in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $?
I'm getting muddled.
Thanks
What is the general form of elements in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $?
I'm getting muddled.
Thanks
In general, if $\alpha$ is an algebraic integer (satisfies a monic polynomial with integer coefficients), and its monic irreducible is of degree $n$, then the elements of $\mathbb{Z}[\alpha]$ can be written uniquely as $a_0 + a_1\alpha + \cdots + a_{n-1}\alpha^{n-1},\qquad a_i\in\mathbb{Z}.$ (A simple application of the remainder theorem for polynomials, and the fact that $\mathbb{Z}[\alpha] = {p(\alpha)\mid p(x)\in\mathbb{Z}[x]}$).
So here, the simplest is to note that $\frac{1+\sqrt{-3}}{2}$ satisfies a monic polynomial of degree $2$ over $\mathbb{Z}$.
In this particular case, there is an alternative possible description: every element of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ can be written uniquely as $\frac{a + b\sqrt{-3}}{2},\qquad a,b\in\mathbb{Z},\qquad a\equiv b\pmod{2}.$