A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros of a cubic polynomial in the expression (with being a third root of unity) only yields two values.
Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation for a prime and, again using modern language, the resulting cyclic Galois group.
These gaps were filled by Niels Henrik Abel in 1824.
Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions.
Évariste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as Galois theory today.
Prior to this, examples of transcendental numbers were known since Joseph Liouville's work in 1844, until Charles Hermite (1873) and Ferdinand von Lindemann (1882) proved the transcendence of and , respectively. The first clear definition of an abstract field is due to .
Both Abel and Galois worked with what is today called an algebraic number field, but conceived neither an explicit notion of a field, nor of a group. In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity).
The English term "field" was introduced by . In 1881 Leopold Kronecker defined what he called a domain of rationality, which is a field of rational fractions in modern terms.
Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem. ==Constructing fields== ===Constructing fields from rings=== A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses .
Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem. ==Constructing fields== ===Constructing fields from rings=== A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses .
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