Definition·D068
Field
A commutative ring with unity, 1 different from 0, in which every nonzero element has a multiplicative inverse.
A field is a commutative ring with unity
in which every nonzero element has a multiplicative inverse: for all
,
In words
A field is a commutative ring with unity where one and zero are different: for every element x, if x is not zero then there is a y in the field with x times y equal to one.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
The rationals are the prototypical example. The requirement
rules out the trivial one-element ring (where
and the inverse condition holds vacuously) from counting as a field.