Applications for $\mathbb Q(\sqrt 2)$

Question

While it's certainly interesting that we can extend fields in surprising ways, are the "first example" type field extensions actually useful for anything? In particular, what about the field $$\Bbb Q[\sqrt{2}] = \{a+b\sqrt{2} \mid a,b \in \Bbb Q\}$$ Does this field help us describe anything or to solve any problems? What's the use of this field?

Answer 1

Let me expand on the comments in a slightly different direction. The field $\Bbb Q(\sqrt2\,)$ and its ring of algebraic integers $\Bbb Z[\sqrt2\,]$ are no more “useful” than any other quadratic field, except for giving the simplest example of a real such field. It’s a principal ideal domain and so has unique factorization, and it (like the other real quadratic fields) has the interesting property that there are more units than just $\{\pm1\}$. These are the integers of the field whose reciprocals also are integers. I’m sure you noticed in high-school algebra that the reciprocal of $\sqrt2+1$ is $\sqrt2-1$. And it’s an interesting fact that every unit of $\Bbb Z[\sqrt2\,]$ is $\pm(1+\sqrt2)^m$ for some $m$. So you may say that this ring is useful as the simplest number field with infinitely many units — though others may give the honor to $\Bbb Z[\frac{1+\sqrt5}2]$.

The fact that $\Bbb Z[\sqrt2\,]$ has unique factorization is seriously interesting. Lots of other real quadratic rings do, and it’s an open question that infinitely many such have this property. The first real quadratic field without unique factorization in its ring of integers is $\Bbb Q(\sqrt{10}\,)$, where $6$ has the double factorization $2\cdot3=-(2-\sqrt{10}\,)(2+\sqrt{10}\,)$, though it takes just a little work to see that none of these four factors is related to another by a unit multiple.

For me, the moral of the story is that the quadratic number fields and their respective rings of integers give lots of examples of phenomena occurring generally in algebraic number theory, but yet these fields are easy to compute in.