Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave … See more

In this section, F denotes an arbitrary field and a and b are arbitrary elements of F.
Consequences of the definition
One has a ⋅ 0 = 0 … See more

Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry. A first step towards … See more

Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular … See more

Rational numbers
Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers that can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a … See more

Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example F4 is a field with four elements. Its subfield F2 is the smallest field, because by definition a field … See more

Constructing fields from rings
A commutative ring is a set that is equipped with an addition and multiplication operation and satisfes all the axioms of a field, … See more