Let R be the ring of integers in . Determine the units in R.
Let be a unit of R, then the norm of x, that is,
I can only think of +1 and -1 being units in R, anything more?
A unit in a ring with multipicative identity 1 is a divisor of 1; if the ring is also a normed vector space, then the norm of a unit must be 1.
So is a unit in ..
Conversely, any element of the form where is a unit in , its mutliplicative inverse being :
Hence the units of are .
By the way, there are units other than ±1: for example, .