Don't know if this helps, but: It occurred to me that maybe a finite field F(p^n) might possibly live as a subring of a matrix ring over its ground field F(p). Googling led to this paper, "Matrix representation of finite fields": http://www.dtic.mil/dtic/tr/fulltext/u2/a247828.pdf , which finds a generator for a finite fields as a matrix over the ground field. There's also a bunch more interesting stuff here about ways to represent fields (not necessarily visually) at "Topics in normal bases of finite fields": https://arxiv.org/pdf/1304.0420.pdf . —Dan Henry Baker wrote: ----- OK, if I extend the rationals with the root alpha of an irreducible polynomial p[x], I can plot alpha on the complex plane; indeed, I can plot *all* of the roots of p[x] on the complex plane. So all of these "extension roots" live in the complex plane. Is there an analogous (single) place/field where all extension roots of GF(p) live -- i.e., a larger field which includes all of the extension fields of GF(p) ? There seems to be a problem, since there are many (isomorphic) ways to extend GF(p); perhaps these are all different in this larger field? -----