A unit is an element in a ring that has a multiplicative inverse. If 
is an algebraic
integer which divides every algebraic integer
in the field, 
is called a unit in that field.
A given field may contain an infinity of units.
The units of 
are the elements relatively prime to 
. The units in 
which are squares are called
quadratic residues.
All real quadratic fields 
have the two units 
.
The numbers of units in 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 cube root of unity.
 | units of  |
| 1 |  ,  |
| 2 |  |
| 3 |  ,  ,  |
