Finite fields as quotients

Question

Although finite fields are usually introduced as field extensions of fields of prime order, they also arise as quotients of number rings; e.g., $GF(9)$ comes from taking the Gaussian integers mod 3 and $GF(4)$ comes from taking the Eisenstein integers mod 2. It seems likely that mid-19th-century mathematicians would have stumbled across specific finite fields in this setting and perhaps also noticed the connection to the work of Galois (though of course the general concept of a finite field is a modern one). Can anyone steer me toward relevant primary or secondary literature?

Answer 1

In the famous XI-th Supplement in Dirichlet-Dedekind's Vorlesungen über Zahlentheorie, quotient rings modulo an ideal (prime ideal or general nonzero ideal) are not used, but systems of representatives are used, just like the concrete way students may first learn to do arithmetic mod $m$ in the integers using the representatives $0, 1, \ldots, m-1$ exclusively.

On p. 509 (in Section 171), Dedekind writes $(\mathfrak a,\mathfrak b)$ for the number of incongruent numbers in $\mathfrak a$ modulo $\mathfrak b$.

On p. 564 (in Section 180) Dedekind writes $(\mathfrak o,{\mathfrak o}\mu) = \pm{N}(\mu)$, where $N$ is the norm mapping. Today we'd say $\mathfrak o/(\mu)$ has size $|N(\mu)|$. I guess Weierstrass's absolute value symbol was not yet universal.

On pp. 570-572 (still in Section 180), Dedekind describes basic properties of elements of $\mathfrak o$ modulo $\mathfrak p$, which we'd call the finite field $\mathfrak o/\mathfrak p$, e.g., the $p$th power map is additive mod $\mathfrak p$ and for each $\alpha \bmod \mathfrak p$ there's a least $a \geq 1$ such that $\alpha^{p^a} \equiv a \bmod \mathfrak p$, and in terms of $a$ the minimal polynomial of $\alpha \bmod \mathfrak p$ over $\mathbf Z/p\mathbf Z$ is $(t-\alpha)(t-\alpha^p)\cdots (t-\alpha^{p^{a-1}})$.

In footnotes on these pages about finite fields arising as number field quotient rings $\mathfrak o/\mathfrak p$, Dedekind refers the reader to his earlier papers Ueber die Discriminanten endlicher Körper (the term "finite field" in that title means number fields, as they have finite dimension over the rationals) and Ueber den Zusammenhang zwischen der Theorie der ideale und der Theorie der höheren Congruenzen. The second paper has been translated into English by Gouvea and Webster: see here.

In those days, people spoke about a "higher congruence" (see the end of the 2nd title) or "double modulus". A higher congruence was a congruence between polynomials whose coefficients are themselves in a field defined by congruences, namely $\mathbf F_p[x]/(f(x))$. For example, they might view our $\mathbf Z[i]/(3)$ or $(\mathbf Z/3\mathbf Z)[x]/(x^2+1)$ as congruences mod $3,x^2+1$. The earlier post on this site about irreducible polynomials is worth reading.

Dickson, in the preface to his 1900 book "Linear groups with an exposition of the Galois field theory" (note "Galois field theory" = finite field theory), wrote

The linear groups investigated by Galois, Jordan and Serret were defined for the field of integers taken modulo $p$; the general Galois field enters only incidentally in their investiagations. The linear fractional group in a general Galois field was partially investigated by Mathieu, and exhaustively by Moore, Burnside and Wiman. The work of Moore first emphasized the importance of employing in group problems the general Galois field in place of the special field of integers [modulo $p$], the results being almost as simple and the investigations no more complicated. In this way the systems of linear groups studied by Jordan have all be [sic] generalized by the author and in the investigation of new systems the Galois field has been employed ab initio.

I had never heard of Wiman before. It is Anders Wiman. In any case, Burnside, Dickson, Moore, and Wiman all began their work involving finite fields in the 1890s (Burnside had earlier worked in applied math), so I think the viewpoint that "everything we can be do for the integers mod $p$ should be done for all finite fields" really took off in that decade and the early years of the 20th century, not during the mid-19th century. This isn't a surprise, since the classification of general finite fields is due to Moore in 1893.