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

, $\pm 1$ , $\pm 2$ , .... The following table gives the fundamental units for the first few real quadratic fields.

	$\eta(d)$		$\eta(d)$
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	$48+7\sqrt{47}$

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.