General form of elements in $ \mathbb{Z} [\frac{1+\sqrt{-3}}{2} ] $

What is the basic kind of components in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $?

I'm obtaining jumbled.

Many thanks

2022-06-07 14:32:13
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Answers: 1

In basic, if $\alpha$ is an algebraic integer (satisfies a monic polynomial with integer coefficients), and also its monic irreducible is of level $n$, after that the components of $\mathbb{Z}[\alpha]$ can be created distinctly as $$a_0 + a_1\alpha + \cdots + a_{n-1}\alpha^{n-1},\qquad a_i\in\mathbb{Z}.$$ (A straightforward application of the rest theory for polynomials, and also the reality that $\mathbb{Z}[\alpha] = {p(\alpha)\mid p(x)\in\mathbb{Z}[x]}$).

So below, the most basic is to keep in mind that $\frac{1+\sqrt{-3}}{2}$ pleases a monic polynomial of level $2$ over $\mathbb{Z}$.

In this certain instance, there is a different feasible summary: every component of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ can be created distinctly as $$\frac{a + b\sqrt{-3}}{2},\qquad a,b\in\mathbb{Z},\qquad a\equiv b\pmod{2}.$$

2022-06-07 15:06:49