My understanding is that the term "field" in science was first used in physics, while the mathematical term, at least the algebraic one, was more recent. Does anybody know when the first occurrence of the term appeared, and if there was any connection with the physical meaning?
See Earliest Known Uses of Some of the Words of Mathematics, sub voce : Field :
The term Zahlenkörper (body of numbers) is due to Richard Dedekind (1831-1916) (Kline, page 1146). Dedekind used the term in his lectures of 1858 but the term did not come into general use until the early 1890s.
Eliakim Hastings Moore (1862-1932) was apparently the first person to use the English word field in its modern sense and the first to allow for a finite field. He coined the expressions "field of order $s$" and "Galois-field of order $s = q^n$." These expressions appeared in print in December 1893 in the Bulletin of the New York Mathematical Society III.75. The paper was presented to the Congress of Mathematics at Chicago on Aug.25, 1893.
At any event, a decade later Edward Vermilye Huntington [in 1904] wrote:
Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The word field is the English equivalent for DEDEKIND’s term Körper; KRONECKER’s term Rationalitätsbereich, which is often used as a synonym, had originally a somewhat different meaning.
It does not seem to me that there is indications of a relation with Maxwell's concept of the electromagnetic field ...