How is this defined?

Question

Just a short question:

How is the set $\mathbb{Z}[\sqrt5]$ defined ? I thought this is the same as $(\sqrt5)\mathbb{Z}$, but that doesn't make sense.

Answer 1

In ring theory we usually define $\Bbb Z[c]$ to be the set of all polynomial expressions in $c$ with integer coefficients, that is all $a_0+a_1c+a_2c^2+\cdots+a_rc^r$ with $a_0,\ldots,a_r\in\Bbb Z$. Here since $(\sqrt 5)^2=5\in\Bbb Z$ we have $$\Bbb Z[\sqrt5]=\{a+b\sqrt 5:a,b\in\Bbb Z\}.$$

Answer 2

This is a good question that isn't talked about too much usually. The way to define $\mathbb{Z}[\sqrt{5}]$ is by considering $$\mathbb{Z}\hookrightarrow \mathbb{Q}\hookrightarrow \overline{\mathbb{Q}}$$ for some fixed $\overline{\mathbb{Q}}$ (algebraic closure is defined up to isomorphism, so you fix one and work with that one for the rest of your book). Then define $\mathbb{Z}[\sqrt{5}]$ to be the intersection of all subrings of $\overline{\mathbb{Q}}$ containing $\sqrt{5}$ (any subring will always contain $\mathbb{Z}$).

Alternatively you can define it as $\mathbb{Z}[x]/(x^2-5)$