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.

The fields with d ≦ 5565 can be classified simply as follows:

{ d ∤ d_{E} \{ h(K_{gen}) = \frac{h(K)}{[K_{gen}:K]} \{ \begin{matrix}h(K) = 1 \\ h(K) > 1 \end{matrix} h(K_{gen}) > \frac{h(K)}{[K_{gen}:K]}  \begin{matrix}⋯ & K_{ur} = K \\ ⋯ & K_{ur} = K_{1} \\ ⋯ & K_{ur} = K_{2} \end{matrix} \}K_{ur} = (K_{gen})_{1}

d | d_{E} \{ d = d_{E} \{ \begin{matrix}prime \\ composite \end{matrix} d = d′d_{E} (d′ fundamental quadratic discriminant) \begin{matrix}⋯ & K_{ur} = K_{3} \\ ⋯ & K_{ur} = K_{2} \\ ⋯ & K_{ur} = K_{2} \end{matrix} \}K_{ur} ⫌ (K_{gen})_{1}

Here, we denote by K_{gen} the wide genus field of K and by (K_{gen})_{1} its Hilbert class field. We mean by ‘d_{E} | d’ that the discriminant d of K is divisible by the discriminant of some quartic number field E.

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

\begin{matrix}d & Cl(K) & l & G & d & Cl(K) & l & G \\ 520 & V_{4} & 2 & Q_{8} & 2652 & V_{4} & 2 & D_{4} \\ 1145 & C_{4} & 2 & Q_{12} & 2748 & C_{2} & 2 & D_{3} \\ 1160 & V_{4} & 2 & D_{4} & 2777 & C_{3} & 3 & Ã_{4} \\ 1285 & C_{2} & 2 & D_{3} & 2840 & C_{2} & 2 & D_{3} \\ 1580 & C_{2} & 2 & D_{3} & 2856 & V_{4} & 2 & D_{4} \\ 1605 & C_{2} & 2 & D_{3} & 2885 & C_{2} & 2 & D_{7} \\ 1640 & V_{4} & 2 & D_{4} & 2977 & C_{2} & 2 & D_{3} \\ 1740 & V_{4} & 2 & Q_{16} & 3045 & V_{4} & 2 & D_{4} \\ 1785 & C_{4} × C_{2} & 2 & M_{16} & 3084 & C_{2} & 2 & D_{3} \\ 1820 & V_{4} & 2 & Q_{8} & 3145 & V_{4} & 2 & Q_{8} \\ 1832 & C_{2} & 2 & D_{3} & 3180 & V_{4} & 2 & Q_{8} \\ 1885 & V_{4} & 2 & D_{4} & 3208 & C_{2} & 2 & D_{5} \\ 1957 & C_{3} & 2 & A_{4} & 3341 & C_{4} & 2 & Q_{12} \\ 2005 & C_{4} & 2 & Q_{20} & 3445 & V_{4} & 2 & D_{4} \\ 2056 & C_{4} & 2 & Q_{12} & 3480 & C_{4} × C_{2} & 2 & C_{4} ⋉ C_{4} \\ 2120 & V_{4} & 2 & Q_{8} & 3485 & V_{4} & 2 & D_{6} \\ 2345 & C_{2} & 2 & D_{3} & 3560 & V_{4} & 2 & D_{4} \\ 2405 & V_{4} & 2 & Q_{8} & 3660 & V_{4} & 2 & D_{4} \\ 2460 & V_{4} & 2 & D_{4} & 3665 & C_{2} & 2 & D_{3} \\ 2465 & V_{4} & 2 & Q_{16} & 3720 & V_{4} & 2 & Q_{8} \\ 2568 & C_{2} & 2 & D_{3} & 3752 & C_{2} & 2 & D_{3} \end{matrix}

Continued (under GRH)

d Cl(K) l  G  d Cl(K) l  G
3784  C_{2} 2 D_{3}  4641 V_{4} × C_{3} 2 D_{4} × C_{3}
3805  C_{4} 2 Q_{12}  4745  V_{4} 2 Q_{8}
3885  V_{4}  2 D_{4}  4785  V_{4} 2 D_{4}
3893  C_{2} 2 D_{3}  4809  C_{2} 2 D_{3}
3965  V_{4}  2 SD_{16}  4812  C_{2} 2 D_{5}
3981  C_{3} 2 A_{4}  4872  V_{4} 2 Q_{8}
4040  V_{4}  2 D_{4}  4920  V_{4} 2 D_{4}
4060  V_{4}  2 D_{4}  4965  C_{2} 2 D_{3}
4085  C_{2} 2 D_{5}  5045  C_{4} 2 Q_{28}
4108  C_{2} 2 D_{3}  5080  C_{2} 2 D_{3}
4173  C_{4} 2 Q_{12}  5213  C_{2} 2 D_{5}
4264  V_{4}  2 D_{4}  5304  V_{4} 2 D_{4}
4305 C_{4} × C_{2} 2 C_{4} ⋉ C_{4} 5340  V_{4} 2 D_{4}
4369  C_{2} 2 D_{3}  5365  V_{4} 2 D_{4}
4460  C_{2} 2 D_{3}  5372  C_{2} 2 D_{3}
4485  V_{4}  2 D_{4}  5465  C_{2} 2 D_{5}
4488 C_{4} × C_{2} 2 C_{4} ⋉ C_{4} 5480  V_{4} 2 D_{4}
4505  V_{4}  2 Q_{16}  5496  C_{2} 2 D_{3}
4520  V_{4}  2 Q_{32}  5397  C_{2} 3  S_{4}
4524  V_{4}  2 D_{4}  5505  C_{2} 2 D_{3}
4616  C_{4} 2 Q_{28}  5512  V_{4} 2 D_{4}
4620  C_{2}^{3} 2 Q_{8} × C_{2}

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 wide 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. Moreover, the unramified A_{4}-extension of Q(\sqrt{2777}) can be embedded into an unramified Ã_{4}-extension of Q(\sqrt{2777}) which is an S̃_{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.