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Field Axioms

The field axioms are conditions on two binary operations, addition and multiplication, on a set F. The additive and multiplicative identity elements are required to satisfy 0!=1. The following laws hold for all a,b,c in F. The inverses shown are required to exist in F, with a multiplicative inverse required only for a!=0.

nameadditionmultiplication
associativity(a+b)+c=a+(b+c)(ab)c=a(bc)
commutativitya+b=b+aab=ba
distributivitya(b+c)=ab+ac(a+b)c=ac+bc
identitya+0=a=0+aa·1=a=1·a
inversesa+(-a)=0=(-a)+aaa^(-1)=1=a^(-1)a if a!=0

See also

Algebra, Field

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References

Apostol, T. M. "The Field Axioms." §I 3.2 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 17-19, 1967.

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Field Axioms

Cite this as:

Weisstein, Eric W. "Field Axioms." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FieldAxioms.html

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