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An element of a set is a unit if its multiplicative inverse belongs to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm 1 and are thus complex roots of unity.
Units in quadratic integer rings
Imaginary quadratic integer rings have only a few units (see: Quadratic integer rings§Theorem I1). The primitive root of unity of degree 3

is used in the following table.
Units of imaginary quadratic integer rings

Units

≤ − 5

{( − 1) 0, ( − 1) 1} = {1, − 1} 

− 3

{ω 0, ω 1, ω 2, ω 3, ω 4, ω 5} = {1, ω, ω 2, − 1, − ω, − ω 2} 

− 2

{( − 1) 0, ( − 1) 1} = {1, − 1} 

− 1

{i 0, i 1, i 2, i 3} = {1, i, − 1, − i} 

Real quadratic integer rings have infinitely many units (see: Quadratic integer rings§Theorem R1), which are all powers of each other, some multiplied by 1. For that reason, the table below can’t be complete like the table above.
An inefficient way to find a unit of a real quadratic integer ring
other than
1 or
− 1 is to try each positive value of
starting with
1 and going up until
is an integer.
Units of real quadratic integer rings

Units

2


3


4

N/A

5

(golden ratio)

6


7


8

N/A, but note that

9

N/A

10


11


12

N/A, but

13


14


15


16

N/A

17





Units

18

N/A, but

19


20

N/A, but

21


22


23


24

N/A, but

25

N/A

26


27

N/A, but

28

N/A, but

29


30


31


32

N/A, but

33



Examples
 The units of are
{i 0, i 1, i 2, i 3} = {1, i, − 1, − i} 
, where . (See Gaussian integers.)
 The units of are .
 The units of are .
 The units of are .
See also