Contrary to simple transcendental extensions of $\mathbb{Q}$, which are necessarily isomorphic to a field of rational functions over $\mathbb{Q}$, simple algebraic extensions are very varied - which explains why we devote so much attention to them during Galois theory.
One particular "family" of such extensions are the cyclotomic extensions (i. e., extensions of the form $\mathbb{Q}(\theta)$, where $\theta^n = 1$ for some $n$). These are quite frequent and I've seen multiple books devoting at least a section specific to them (especially considering prime values of $n$).
That said, this is the only well-known "family" of algebraic numbers I know of (at least, that I can recall). Of course, I could simply give roots of equations of the form, say, $x^n - x - 1$ a name and study their properties, but why focus on this group specifically? This leads me to ponder:
Are there any other groups of algebraic numbers - perhaps roots of equations of a special form - that receive distinguished attention, besides the roots of unity? If so, which ones and where do they show up?
Thanks in advance!
