Let K = Q(\sqrt{d}) be a real quadratic number field with discriminant d and K_{ur} its maximal unramified extension. We denote by G the Galois group Gal(K_{ur}/K) and l be the length of the class field tower of K: K = K_{0} ⫅ K_{1} ⫅ K_{2} ⫅ ⋯ (K_{i+1} is the Hilbert class field of K_{i}), that is, l is the smallest number with K_{l+1} = K_{l}.

Under GRH we have determined the structure of G for d ≦ 5565. For d ≦ 2636, the results are unconditional. We give a table of G for K with K_{ur} ≠ K_{1} below.

We the following notations in the table. In the simple expressions of K_{1} and K_{2}, a_{i}, b_{i} and c_{i} denote any algebraic numbers generating the ith totally real cubic number field, the ith totally real quartic number field with Galois group isomorphic to S_{4}, and the ith totally real quintic number field with Galois group isomorphic to D_{5}, respectively, where we consider that the number fields of each signature and each type (of Galois group of normal closure) are numbered up to conjugacy by absolute values of discriminants. (Here we do not need to consider nonisomorphic fields with same discriminants.)

Table of real quadratic number fields K = Q(\sqrt{d}), d ≦ 5565 with K_{ur} ≠ K_{1}

d Cl(K)  K_{1}  K_{2}  l G
520  V_{4}  K(\sqrt{2}, \sqrt{5})  K(\sqrt{(1 + \sqrt{2})(3\sqrt{2} + \sqrt{5})})  2 Q_{8}
1145  C_{4} K(\sqrt{(31 + 3\sqrt{5})/2})  K_{1}(a_{2})  2 Q_{12}
1160  V_{4}  K(\sqrt{5}, \sqrt{29})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
1285  C_{2} K(\sqrt{5})  K_{1}(a_{3})  2 D_{3}
1580  C_{2} K(\sqrt{5})  K_{1}(a_{4})  2 D_{3}
1605  C_{2} K(\sqrt{5})  K_{1}(a_{5})  2 D_{3}
1640  V_{4}  K(\sqrt{2}, \sqrt{5})  K_{1}(\sqrt{7 + 2\sqrt{2}})  2 D_{4}
1740  V_{4}  K(\sqrt{3}, \sqrt{5})  2 Q_{16}
1785 C_{4} × C_{2} K(\sqrt{5}, \sqrt{(19 + \sqrt{21})/2})  2 M_{16}
1820  V_{4}  K(\sqrt{5}, \sqrt{13})  K(\sqrt{(\sqrt{13} + 2\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8}
1832  C_{2} K(\sqrt{2})  K_{1}(a_{2})  2 D_{3}
1885  V_{4}  K(\sqrt{5}, \sqrt{13})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
1957  C_{3} K(a_{54})  K_{1}(b_{1})  2 A_{4}
2005  C_{4} K(\sqrt{(43 + 7\sqrt{5})/2})  K_{1}(c_{1})  2 Q_{20}
2056  C_{4} K(\sqrt{17 + 4\sqrt{2}})  K_{1}(a_{3})  2 Q_{12}
2120  V_{4}  K(\sqrt{2}, \sqrt{5})  K(\sqrt{(3\sqrt{5} + 7\sqrt{2})(1 + \sqrt{2})}) 2 Q_{8}
2345  C_{2} K(\sqrt{5})  K_{1}(a_{7})  2 D_{3}
2405  V_{4}  K(\sqrt{5}, \sqrt{13})  K(\sqrt{(4\sqrt{13} + 7\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8}
2460  V_{4}  K(\sqrt{3}, \sqrt{5})  K_{1}(\sqrt{(13 + \sqrt{5})/2})  2 D_{4}
2465  V_{4}  K(\sqrt{5}, \sqrt{17})  2 Q_{16}
2568  C_{2} K(\sqrt{2})  K_{1}(a_{5})  2 D_{3}

Typeset by AMS-TEX

Continued (under GRH)

d Cl(K)  K_{1}  K_{2}  l  G
2652  V_{4}  K(\sqrt{3}, \sqrt{13})  K_{1}(\sqrt{(9 + \sqrt{13})/2})  2 D_{4}
2748  C_{2} K(\sqrt{3})  K_{1}(a_{2})  2 D_{3}
2777  C_{3} K(a_{80})  K_{1}(b_{2})  3 Ã_{4}
2840  C_{2} K(\sqrt{5})  K_{1}(a_{10})  2 D_{3}
2856  V_{4}  K(\sqrt{2}, \sqrt{17})  K_{1}(\sqrt{3(5 + 2\sqrt{2})})  2 D_{4}
2885  C_{2} K(\sqrt{5})  2 D_{7}
2977  C_{2} K(\sqrt{13})  K_{1}(a_{2})  2 D_{3}
3045  V_{4}  K(\sqrt{5}, \sqrt{21})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
3084  C_{2} K(\sqrt{3})  K_{1}(a_{3})  2 D_{3}
3145  V_{4}  K(\sqrt{5}, \sqrt{17})  K(\sqrt{(4\sqrt{17} + 11\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8}
3180  V_{4}  K(\sqrt{3}, \sqrt{5})  K(\sqrt{(\sqrt{53} + 4\sqrt{5})(2 + \sqrt{5})})  2 Q_{8}
3208  C_{2} K(\sqrt{2})  K_{1}(c_{1})  2 D_{5}
3341  C_{4} K(\sqrt{33 + 8\sqrt{13}})  K_{1}(a_{3})  2 Q_{12}
3445  V_{4}  K(\sqrt{5}, \sqrt{13})  K_{1}(\sqrt{23 + 2\sqrt{13}})  2 D_{4}
3480 C_{4} × C_{2} K(\sqrt{5}, \sqrt{19 + 6\sqrt{6}})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 C_{4} ⋉ C_{4}
3485  V_{4}  K(\sqrt{5}, \sqrt{17})  K_{1}(a_{12})  2 D_{6}
3560  V_{4}  K(\sqrt{2}, \sqrt{5})  K_{1}(\sqrt{13 + 4\sqrt{5}})  2 D_{4}
3660  V_{4}  K(\sqrt{3}, \sqrt{5})  K_{1}(\sqrt{3(9 + 2\sqrt{5})})  2 D_{4}
3665  C_{2} K(\sqrt{5})  K_{1}(a_{23})  2 D_{3}
3720  V_{4}  K(\sqrt{2}, \sqrt{5})  K(\sqrt{(7\sqrt{2} + \sqrt{5})(1 + \sqrt{2})})  2 Q_{8}
3752  C_{2} K(\sqrt{2})  K_{1}(a_{7})  2 D_{3}
3784  C_{2} K(\sqrt{2})  K_{1}(a_{8})  2 D_{3}
3805  C_{4} K(\sqrt{29 + 4\sqrt{5}})  K_{1}(a_{7})  2 Q_{12}
3885  V_{4}  K(\sqrt{5}, \sqrt{21})  K_{1}(\sqrt{(11 + \sqrt{37})/2})  2 D_{4}
3893  C_{2} K(\sqrt{17})  K_{1}(a_{2})  2 D_{3}
3965  V_{4}  K(\sqrt{5}, \sqrt{13})  2 SD_{16}
3981  C_{3} K(a_{127})  K_{1}(b_{3})  2 A_{4}
4040  V_{4}  K(\sqrt{2}, \sqrt{5})  K_{1}(\sqrt{11 + 2\sqrt{5}})  2 D_{4}
4060  V_{4}  K(\sqrt{5}, \sqrt{7})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
4085  C_{2} K(\sqrt{5})  K_{1}(c_{2})  2 D_{5}
4108  C_{2} K(\sqrt{13})  K_{1}(a_{4})  2 D_{3}
4173  C_{4} K(\sqrt{(59 + 13\sqrt{13})/2})  K_{1}(a_{5})  2 Q_{12}
4264  V_{4}  K(\sqrt{2}, \sqrt{13})  K_{1}(\sqrt{7 + 2\sqrt{2}})  2 D_{4}
4305 C_{4} × C_{2} K(\sqrt{5}, \sqrt{(43 + 7\sqrt{21})/2}) K_{1}(\sqrt{3(13 + \sqrt{5})/2})  2 C_{4} ⋉ C_{4}
4369  C_{2} K(\sqrt{17})  K_{1}(a_{3})  2 D_{3}
4460  C_{2} K(\sqrt{5})  K_{1}(a_{19})  2 D_{3}

Continued (under GRH)

d Cl(K)  K_{1}  K_{2}  l  G
4485  V_{4}  K(\sqrt{5}, \sqrt{13})  K_{1}(\sqrt{11 + 2\sqrt{13}})  2 D_{4}
4488 C_{4} × C_{2} K(\sqrt{2}, \sqrt{(31 + 5\sqrt{33})/2}) K_{1}(\sqrt{3(5 + 2\sqrt{2})})  2 C_{4} ⋉ C_{4}
4505  V_{4}  K(\sqrt{5}, \sqrt{17})  2 Q_{16}
4520  V_{4}  K(\sqrt{2}, \sqrt{5})  2 Q_{32}
4524  V_{4}  K(\sqrt{3}, \sqrt{13})  K_{1}(\sqrt{9 + 2\sqrt{13}})  2 D_{4}
4616  C_{4} K(\sqrt{33 + 16\sqrt{2}})  2 Q_{28}
4620  C_{2}^{3} K(\sqrt{3}, \sqrt{5}, \sqrt{7})  K_{1}(\sqrt{(4\sqrt{3} + 5\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8} × C_{2}
4641 V_{4} × C_{3} K(\sqrt{13}, \sqrt{17}, a_{146})  K_{1}(\sqrt{3(9 + \sqrt{13})/2})  2 D_{4} × C_{3}
4745  V_{4}  K(\sqrt{5}, \sqrt{13})  K(\sqrt{(2\sqrt{13} + 5\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8}
4785  V_{4}  K(\sqrt{5}, \sqrt{29})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
4809  C_{2} K(\sqrt{21})  K_{1}(a_{2})  2 D_{3}
4812  C_{2} K(\sqrt{3})  K_{1}(c_{1})  2 D_{5}
4872  V_{4}  K(\sqrt{2}, \sqrt{21})  K(\sqrt{(\sqrt{21} + 5\sqrt{29})(5 + \sqrt{29})}) 2 Q_{8}
4920  V_{4}  K(\sqrt{5}, \sqrt{6})  K_{1}(\sqrt{3(11 + 4\sqrt{5})})  2 D_{4}
4965  C_{2} K(\sqrt{5})  K_{1}(a_{22})  2 D_{3}
5045  C_{4} K(\sqrt{33 + 4\sqrt{5}})  2 Q_{28}
5080  C_{2} K(\sqrt{5})  K_{1}(a_{23})  2 D_{3}
5213  C_{2} K(\sqrt{13})  K_{1}(c_{1})  2 D_{5}
5304  V_{4}  K(\sqrt{6}, \sqrt{13})  K_{1}(\sqrt{3(9 + \sqrt{13})/2})  2 D_{4}
5340  V_{4}  K(\sqrt{3}, \sqrt{5})  K_{1}(\sqrt{13 + 4\sqrt{5}})  2 D_{4}
5365  V_{4}  K(\sqrt{5}, \sqrt{29})  K_{1}(\sqrt{7 + 2\sqrt{5}})  2 D_{4}
5372  C_{2} K(\sqrt{17})  K_{1}(a_{4})  2 D_{3}
5465  C_{2} K(\sqrt{5})  K_{1}(c_{3})  2 D_{5}
5480  V_{4}  K(\sqrt{2}, \sqrt{5})  K_{1}(\sqrt{13 + 4\sqrt{2}})  2 D_{4}
5496  C_{2} K(\sqrt{6})  K_{1}(a_{2})  2 D_{3}
5397  C_{2} K(\sqrt{21})  K_{1}(a_{3})  3  S_{4}
5505  C_{2} K(\sqrt{5})  K_{1}(a_{25})  2 D_{3}
5512  V_{4}  K(\sqrt{2}, \sqrt{13})  K_{1}(\sqrt{(15 + \sqrt{13})/2})  2 D_{4}

As a remarkable fact, we note the following. Except only for the four fields Q(\sqrt{1957}), Q(\sqrt{2777}), Q(\sqrt{3981}), Q(\sqrt{5397}), K_{ur} coincides with the Hilbert class field of the genus field of K for d ≦ 5565. The fields Q(\sqrt{1957}), Q(\sqrt{2777}), Q(\sqrt{3981}) have an unramified A_{4}-extension which is an S_{4}-extension of Q. The field Q(\sqrt{5397}) has an unramified S_{4}-extension which is compositum with an S_{4}-extension of Q.