A unit is an element 
of a ring with multiplicative
identity 1 for which an element 
satisfies 
.
Every nonzero element of
a field is a unit. For a number
field 
, the units of 
considered here are the units of its ring
of integers 
.
Equivalently, they are the algebraic integers
in 
that divide 1 in 
, and hence divide every algebraic
integer in 
.
The ring 
may have infinitely many units.
For an integer 
, the units of 
are the residue classes
represented by integers relatively
prime to 
.
Among these units, the squares modulo 
are quadratic residues.
All real quadratic fields 
have 
as units of their rings
of integers.
The numbers of units in the ring of integers of the imaginary quadratic field 
for 
, 2, ... are 4, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 6, 2, ... (OEIS
A092205). There are four units for 
, 4, 9, 16, ... (OEIS A000290;
the square numbers), six units for 
, 12, 27, 48, ... (OEIS A033428;
three times the square numbers), and two units for
all other imaginary quadratic fields, i.e., 
, 5, 6, 7, 8, 10, 11, ... (OEIS A092206).
The following table gives the units for small 
. In this table, 
is a primitive root
of unity of order 3.
 | units of  |
| 1 |  ,  |
| 2 |  |
| 3 |  ,  ,  |
