# Field (mathematics)

### 1770

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.

### 1824

These gaps were filled by Niels Henrik Abel in 1824.

### 1830

Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions.

### 1832

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.

### 1844

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 .

### 1871

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).

### 1881

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.

### 1928

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 .

### 1942

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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