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 similarly as they behave for

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Rational numbers
Rational numbers have been widely used a long time before the elaboration of the concept of field. They are

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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 and −a = (−1) ⋅ a. In particular, one may deduce the additive inverse of every element as soon as one knows −1.

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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 has at

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Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory,

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Constructing fields from rings
A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms

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Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.
Ordered fields
A field F is called an ordered field if any two elements can be

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