W. H. Bussey (1905) "Galois field tables for $p^n \le 169$", Bulletin of the American Mathematical Society 12(1): 22–38, doi:10.1090/S0002-9904-1905-01284-2
- When were finite fields introduced?
- And why?
- When were they first called Galois fields?
- Is the name Galois field justified?
Galois introduced finite fields in this paper:
Galois, Évariste (1830). "Sur la théorie des nombres". Bulletin des Sciences mathématiques. XIII: 428.
So says Wikipedia, and also Dickson in his 1900 book on the topic.
EDIT: There's more information in Lidl and Niederreiter, Finite Fields (Volume 20 in the Encyclopedia of Mathematics series). For the Galois paper, they give the following bibliographical information:
Bull. Sci. Math. de M. Férussac 13 (1830) 428-435; J. Math. Pures Appl. 11 (1846) 398-407; Oeuvres math., pp. 15-23, Gauthier-Villars, Paris, 1897.
They also say, "Rudiments of the theory are also contained in a posthumous manuscript of Gauss" Analysis residuorum: Caput octavum. Disquisitiones generales de congruentiis, Werke, vol. 2, pp. 212-240, Konigl. Gesellschaft der Wissenschaften, Gottingen, 1876; Untersuchungen uber Hohere Arithmetik (H. Maser, ed.), pp. 630-652, Springer, Berlin, 1889.
And they say see Niederreiter, Richard Dedekind and the development of the theory of finite fields, Abh. Braunschweig. Wissenschaftl. Gesellschaft 33 (1982) 183-187, for a discussion of the early history of the subject.