This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help.

Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{F}_{q}$. Is there a clean reference in the literature to the fact that there must be at least one regular semisimple element in $G(K)$?

I know from the literature that regular semisimple elements are dense in $G$ and in every maximal torus $T$ (for example §2.3 and §2.5 in Humphreys's *Conjugacy classes in semisimple algebraic groups*), but this ensures only that there are many elements over the algebraic closure $\overline{K}$, not specifically over $K$. I know that this is true for $\mathrm{char}(K)=0$, using unirationality and the fact that $K$ is infinite, as in this question. I even know how to presumably shoot my question with a cannon in a few different ways, for instance:

- Simply connected (almost) simple groups are in 1-1 correspondence with Dynkin diagrams (see for instance §32 in Humphreys's
*Linear algebraic groups*), so I could manually write down one regular semisimple element over $\mathbb{F}_{q}$ for each such $G$. [EDIT: as pointed out in the comments below by LSpice, since the field in not algebraically closed there's no 1-1 correspondence as stated. One needs to consider automorphisms of the Dynkin diagrams as well.] - Fleischmann and Janiszczak (1993) have formulas for the number of regular semisimple elements over $\mathbb{F}_{q}$ at least for the classical $G$, i.e. of type $A,B,C,D$; they even announce on p. 484 that they obtained the number of semisimple conjugacy classes for $E$, so possibly they cover those cases as well. In any case, if there is a formula and its result is $>0$, then in particular there is one such element.
- Lehrer (1992) showed among other things the "curious" result that, independently of characteristic, any semisimple simply connected $G$ has an odd number of regular semisimple conjugacy classes in $G(K)$ (see Corollary 3.5). Again, in particular if the number is odd then it is not $0$.

I would classify each of the routes above as "too strong for my question". I feel there must be some chapter-1-of-a-textbook kind of reference for "there exists a regular semisimple element over $\mathbb{F}_{q}$", but I could not find one.

Do you know where to find it? Also, bonus content: is the fact true and referenced if I relax the hypotheses (connected, or semisimple, or reductive...)?

class(i.e., the elements of the class might be permuted, but the class itself is fixed, by Galois), then you can find a regular semisimpleelement. $\endgroup$5more comments