History
The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree five or higher.
In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axioms of a field. This construction has been frequently recalled as a contribution to the foundations of mathematics.
In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity), hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert. In 1893, Eliakim Hastings Moore called the concept "field" in English.
In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field. In 1910, Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields). In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension.
Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.
Read more about this topic: Field (mathematics)
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