Does the structure $\mathbb Z [\sqrt m]$ have a name?

Question

It is a popular approach to the study of abstract algebra to introduce the set defined as $\mathbb Z [\sqrt m] := \{a + b \sqrt m: a, b \in \mathbb Z\}$ where $m$ is usually a small prime like $2$ or $3$, but (I believe) could be any number which is not a perfect square.

Then it is set as an exercise to prove that the algebraic structure $(\mathbb Z [\sqrt m], +, \times)$ is an integral domain, and that it is a subdomain of the real numbers $\mathbb R$.

All very straightforward, and can be polished off in a brief few minutes by an attentive and conscientious student.

But the question is: does the structure $\mathbb Z [\sqrt m]$ actually have a name as such?

Some similar structures, involving $i$, have their own names, e.g. $\mathbb Z [i]$ are the Gaussian integers, and $\mathbb Z [\omega]$ where $\omega^3 = 1$ and $\omega \ne 1$ are the Eisenstein integers.

But I am tired of referring to $\mathbb Z [\sqrt m]$ as "the set of numbers of type integer a plus integer b root $m$", and I would dearly love to be able to abbreviate my expositions.

Similarly for $\mathbb Q [\sqrt m]$ where $\mathbb Q$ denotes the rational numbers.

Answer 1

[Edit: I think I slightly mistook the intention of the question. As per Severin Schraven's comment, "$\mathbb Z$ (or $\mathbb Q$) adjoined square root of $m$" is a nice and concise thing to call it in words.]

These things come up a lot in algebraic number theory.

Assume $m$ is squarefree. $\Bbb Q(\sqrt m)$ are called quadratic fields (meaning (algebraic) number field of degree $2$). $\Bbb Z[\sqrt m]$ is an order in $\Bbb Q(\sqrt m)$ (in particular a module and a subring). If $m\not\equiv1\pmod4$, then $\Bbb Z[\sqrt m]$ is in fact the ring of integers of $\Bbb Q(\sqrt m)$ (which is the same as the maximal order). In the case $m\equiv1\pmod4$, the ring of integers is $\Bbb Z\left[\frac{1+\sqrt m}{2}\right]$.

Note that "ring of integers" refers to the notion of algebraic integers (as opposed to your plain old integers in $\Bbb Z$). Meaning they are roots of a monic polynomial with integer coefficients. The ring of integers of a number field $K$ is just the set of algebraic integers in $K$. The elements you find in $\Bbb Z[\sqrt m]$ or $\Bbb Z\left[\frac{1+\sqrt m}{2}\right]$ (according to $m$'s residue class) are called the quadratic integers, because they are exactly the roots of monic quadratic polynomials with integer coeffiecients.

You may see the notation $\mathcal O_K$ for the ring of integers of the number field $K$.

Number fields and their ring of integers are natural things to study in algebraic number theory, so the structures are definitely important. This combined with the fact that they can be easily defined and worked with for beginners explains the fact that they come up as early examples a lot. Oh, and of course they make for a convienient collection of examples and counterexamples when varying $m$. It is a bit of a shame IMO that some textbooks (such as Dummit&Foote at least) don't do anything to motivate them, or at least mention that "these things are actually important". Or give them a name for the curious student to look up.