Dual Number

A dual number over a field is an element of the quotient ring . Every dual number has a unique expression , where and denotes the nonzero class satisfying (Milne 2024, p. 90). Addition is componentwise, and multiplication is given by

Thus is a nonzero nilpotent element, so this ring is not a reduced ring and is not a field.

For , a dual number can be represented by matrices. Take and interpret the scalar as , where is the identity matrix. This gives , with . A dual number is a ring unit iff , in which case its multiplicative inverse is