Fundamental Unit

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, for the maximal such that there is a primitive th root of unity of . Whenever is quadratic, (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. Observe that is the order of the torsion subgroup of and that the are determined up to a change of -basis and up to a multiplication by a root of unity.

The fundamental unit of a number field is intimately connected with the regulator.

The fundamental units of a field generated by the algebraic number can be computed in the Wolfram Language using NumberFieldFundamentalUnits[a].

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 1987, 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.

For nonsquare positive integers, the minimal s are given by 2, 4, 1, 10, 16, 2, 6, 20, 4, 3, 30, 8, 8, 34, 340, 4, 5, ... (OEIS A048941), while the minimal s are given by 2, 2, 1, 4, 6, 1, 2, 6, 1, 1, 8, 2, 2, 8, 78, 1, 1, 84, ... (OEIS A048942). Given a minimal , 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).

	OEIS	squarefree numbers with
1	A107997	5, 13, 21, 29, 53, 61, 69, 77, 85, 93, ...
2	A107998	2, 3, 6, 7, 10, 11, 14, 15, 17, 19, 22, ...