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Fundamental Unit

In a real Quadratic Field, there exists a special Unit $\eta$ known as the fundamental unit such that all units $\rho$ are given by $\rho=\pm \eta^m$, for $m=0$, $\pm 1$, $\pm 2$, .... The following table gives the fundamental units for the first few real quadratic fields.

$d$ $\eta(d)$ $d$ $\eta(d)$
2 $1+\sqrt{2}$ 51 $50+7\sqrt{51}$
3 $2+\sqrt{3}$ 53 ${\textstyle{1\over 2}}(7+\sqrt{53}\,)$
5 ${\textstyle{1\over 2}}(1+\sqrt{5}\,)$ 55 $89+12\sqrt{55}$
6 $5+2\sqrt{6}$ 57 $151+20\sqrt{57}$
7 $8+3\sqrt{7}$ 58 $99+13\sqrt{58}$
10 $3+\sqrt{10}$ 59 $530+69\sqrt{59}$
11 $10+3\sqrt{11}$ 61 ${\textstyle{1\over 2}}(39+5\sqrt{61}\,)$
13 ${\textstyle{1\over 2}}(3+\sqrt{13}\,)$ 62 $63+8\sqrt{62}$
14 $15+4\sqrt{14}$ 65 $8+\sqrt{65}$
15 $4+\sqrt{15}$ 66 $65+8\sqrt{66}$
17 $4+\sqrt{17}$ 67 $48842+5967\sqrt{67}$
19 $170+39\sqrt{19}$ 69 ${\textstyle{1\over 2}}(25+3\sqrt{69}\,)$
21 ${\textstyle{1\over 2}}(5+\sqrt{21}\,)$ 70 $251+30\sqrt{70}$
22 $197+42\sqrt{22}$ 71 $3480+413\sqrt{71}$
23 $24+5\sqrt{23}$ 73 $1068+125\sqrt{73}$
26 $5+\sqrt{26}$ 74 $43+5\sqrt{74}$
29 ${\textstyle{1\over 2}}(5+\sqrt{29}\,)$ 77 ${\textstyle{1\over 2}}(9+\sqrt{77}\,)$
30 $11+2\sqrt{30}$ 78 $53+6\sqrt{78}$
31 $1520+273\sqrt{31}$ 79 $80+9\sqrt{79}$
33 $5+4\sqrt{33}$ 82 $9+\sqrt{82}$
34 $35+6\sqrt{34}$ 83 $82+9\sqrt{83}$
35 $6+\sqrt{35}$ 85 ${\textstyle{1\over 2}}(9+\sqrt{85}\,)$
37 $6+\sqrt{37}$ 86 $10405+1122\sqrt{86}$
38 $37+6\sqrt{38}$ 87 $28+3\sqrt{87}$
39 $25+4\sqrt{39}$ 89 $500+53\sqrt{89}$
41 $32+5\sqrt{41}$ 91 $1574+165\sqrt{91}$
42 $13+2\sqrt{42}$ 93 ${\textstyle{1\over 2}}(29+3\sqrt{93}\,)$
43 $3482+531\sqrt{43}$ 95 $39+4\sqrt{95}$
46 $24335+3588\sqrt{46}$ 97 $5604+569\sqrt{97}$
47 $48+7\sqrt{47}$    

See also Quadratic Field, Unit


References

Cohn, H. ``Fundamental Units'' and ``Construction of Fundamental Units.'' §6.4 and 6.5 in Advanced Number Theory. New York: Dover, pp. 98-102, and 261-274, 1980.

mathematica.gif Weisstein, E. W. ``Class Numbers.'' Mathematica notebook ClassNumbers.m.



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© 1996-9 Eric W. Weisstein
1999-05-26