It is very cool that matrix algebras act as some kind of universal systems in which all lesser algebras sit. —Dan

From: "W. Edwin Clark" <wclark@mail.usf.edu> Sent: Jul 22, 2017 8:29 PM

Finite fields are finite dimensional associative algebras over their ground fields (or any subfield).

In general every n-dimensional associative algebra A over a field F, is isomorphic to a subalgebra of the algebra of all n x n matrices over F if A has an identity--- otherwise A can be embedded in an algebra with identity of dimension n + 1 and thus becomes a subalgebra of (n+1) x (n+1) matrices over F.

On Sat, Jul 22, 2017 at 4:50 PM, Dan Asimov <dasimov@earthlink.net> wrote:

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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

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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? -----