Let be a number
field with real embeddings and imaginary embeddings and let . Then
the multiplicative group of units of has the form
(1)
where is a primitive th root of unity,
(unless , in which
case , or ,
in which case ). Thus, is isomorphic
to the group . The generators for are
called the fundamental units of . Real quadratic
number fields and imaginary cubic number fields have just one fundamental unit and
imaginary quadratic number fields have no fundamental units.
In a real quadratic field, there exists a special unit known as the fundamental
unit such that all units are given by , for , , , .... The notation
is sometimes used instead of (Zucker and Robertson 1976). The fundamental units
for real quadratic fields may be computed from the fundamental solution
of the Pell equation
(2)
where the sign is taken such that the solution has smallest
possible positive (LeVeque 1977; Cohn 1980,
p. 101; Hua 1982; Borwein and Borwein 1986, p. 294). If the positive sign is taken,
then one solution is simply given by ,
where is the solution to the Pell equation
(3)
However, this need not be the minimal solution. For example, the solution
to Pell equation
(4)
is , so ,
but is the minimal solution. Given
a minimal (Sloane's A048941 and A048942), the fundamental unit is given by
(5)
(Cohn 1980, p. 101).
The following table gives fundamental units for small .
2
54
3
55
5
56
6
57
7
58
8
59
10
60
11
61
12
62
13
63
14
65
15
66
17
67
18
68
19
69
20
70
21
71
22
72
23
73
24
74
26
75
27
76
28
77
29
78
30
79
31
80
32
82
33
83
34
84
35
85
37
86
38
87
39
88
40
89
41
90
42
91
43
92
44
93
45
94
46
95
47
96
48
97
50
98
51
99
52
101
53
102
The following table given the squarefree numbers for which the denominator
of is for or 2. These
sequences turn out to be related to Eisenstein's problem: there is no known fast
way to compute them for large (Finch).
be a 
real embeddings and 
imaginary embeddings and let 
. Then
the multiplicative group of units 
of 
has the form
is a primitive 
th root of unity,
(unless 
, in which
case 
, or 
,
in which case 
). Thus, 
is isomorphic
to the group 
. The 
generators 
for 
are
called the fundamental units of 
. Real quadratic
number fields and imaginary cubic number fields have just one fundamental unit and
imaginary quadratic number fields have no fundamental units.
known as the fundamental
unit such that all units 
are given by 
, for 
, 
, 
, .... The notation
is sometimes used instead of 
(Zucker and Robertson 1976). The fundamental units
for 
may be computed from the fundamental solution
of the 
has smallest
possible positive 
(LeVeque 1977; Cohn 1980,
p. 101; Hua 1982; Borwein and Borwein 1986, p. 294). If the positive sign is taken,
then one solution is simply given by 
,
where 
is the solution to the 
, so 
,
but 
is the minimal solution. Given
a minimal 
(Sloane's 
.
for which the denominator
of 
is 
for 
or 2. These
sequences turn out to be related to Eisenstein's problem: there is no known fast
way to compute them for large 
(Finch).
with 