Definition:Field (Abstract Algebra)

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This page is about field in abstract algebra. For other uses, see Definition:Field.

Definition

A field is a non-trivial division ring whose ring product is commutative.


Thus, let $\left({F, +, \times}\right)$ be an algebraic structure.


Then $\left({F, +, \times}\right)$ is a field iff:


This definition gives rise to the field axioms, as follows:


Field Axioms

The properties of a field are as follows.

For a given field $\left({F, +, \circ}\right)$, these statements hold true:

\((A0):\)   Closure under addition      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x + y \in F \)             
\((A1):\)   Associativity of addition      \(\displaystyle \forall x, y, z \in F:\) \(\displaystyle \left({x + y}\right) + z = x + \left({y + z}\right) \)             
\((A2):\)   Commutativity of addition      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x + y = y + x \)             
\((A3):\)   Identity element for addition      \(\displaystyle \exists 0_F \in F: \forall x \in F:\) \(\displaystyle x + 0_F = x = 0_F + x \)             $0_F$ is called the zero
\((A4):\)   Inverse elements for addition      \(\displaystyle \forall x: \exists x' \in F:\) \(\displaystyle x + x' = 0_F = x' + x \)             $x'$ is called a negative element
\((M0):\)   Closure under product      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x \circ y \in F \)             
\((M1):\)   Associativity of product      \(\displaystyle \forall x, y, z \in F:\) \(\displaystyle \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right) \)             
\((M2):\)   Commutativity of product      \(\displaystyle \forall x, y \in F:\) \(\displaystyle x \circ y = y \circ x \)             
\((M3):\)   Identity element for product      \(\displaystyle \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) \(\displaystyle x \circ 1_F = x = 1_F \circ x \)             $1_F$ is called the unity
\((M4):\)   Inverse elements for product      \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x \)             
\((D):\)   Product is distributive over addition      \(\displaystyle \forall x, y, z \in F:\) \(\displaystyle x \circ \left({y + z}\right) = \left({x \circ y}\right) + \left({x \circ z}\right) \)             


These are called the field axioms.


Internationalization

Field is translated:

In Dutch: lichaam  (literally: body)
In French: corps  (literally: body)
In Spanish: cuerpo  (literally: body)


Also see

  • Results about fields can be found here.


Sources