A field is a set F together with two binary operations on F, called addition and multiplication, and denoted + and , satisfying the following properties, for all a,b,cF:

  1. 1.

    a+(b+c)=(a+b)+c (associativity of addition)

  2. 2.

    a+b=b+a (commutativity of addition)

  3. 3.

    a+0=a for some element 0F (existence of zero elementMathworldPlanetmath)

  4. 4.

    a+(-a)=0 for some element -aF (existence of additive inverses)

  5. 5.

    a(bc)=(ab)c (associativity of multiplication)

  6. 6.

    ab=ba (commutativity of multiplication)

  7. 7.

    a1=a for some element 1F, with 10 (existence of unity element)

  8. 8.

    If a0, then aa-1=1 for some element a-1F (existence of multiplicative inversesMathworldPlanetmath)

  9. 9.

    a(b+c)=(ab)+(ac) (distributive property)

Equivalently, a field is a commutative ring F with identityPlanetmathPlanetmath such that:

  • 10

  • If aF, and a0, then there exists bF with ab=1.

Title field
Canonical name Field
Date of creation 2013-03-22 11:48:43
Last modified on 2013-03-22 11:48:43
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 12E99
Classification msc 03A05