Fundamental Unit

The fundamental units of a number field with real embeddings and imaginary embeddings are the generators, where , in the multiplicative group of units of its ring of integers, having the form

(1)

where is a primitive wth 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 fundamental units are indexed by . The number of fundamental units of is therefore . Real quadratic number fields and imaginary cubic number fields have just one fundamental unit and imaginary quadratic number fields have no fundamental units. The integer is the order of the torsion subgroup of , and the are determined up to a change of -basis and up to 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 unit of with squarefree integer may be computed from the fundamental solution of the Pell equation

(2)

where the sign is taken such that the positive integer solution has smallest possible (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). For squarefree , the fundamental unit is

(5)

(Cohn 1980, p. 101). For nonsquarefree , the same integral Pell equation minima need not give the fundamental unit of the real quadratic field. For example, at they give , while the field's fundamental unit is .

The following table gives fundamental units of for small nonsquare , using the squarefree part of .

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 gives the squarefree numbers for which the common denominator of the coefficients in 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
2	A107997	5, 13, 21, 29, 53, 61, 69, 77, 85, 93, ...
1	A107998	2, 3, 6, 7, 10, 11, 14, 15, 17, 19, 22, ...