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 |  ,  ,  |
