Galois action on class groups. (English) Zbl 1032.12008

The goal of this article is to give a unified approach for the study of the constraints placed on the structure of the class group of a normal extension by the action of the Galois group of the extension. For a number field \(K\) and a prime \(p\), let \(\text{Cl}(K)\) and \(\text{CL}_p(K)\) denote the class group of \(K\) and its \(p\)-Sylow subgroup. For a normal extension \(L/F\) of number fields the relative class group is the kernel of the norm mapping \(N_{L/F}: \text{Cl}(L)\to \text{Cl}(F)\). It and its \(p\)-Sylow subgroup will be denoted by \(C(L/F)\) and \(\text{CL}_p(L/F)\). It is first shown that if \(p\) does not divide \([L:F]\) then \(\text{Cl}_p(L)\cong \text{CL}_p(L/F)\times \text{Cl}_p(F)\).
The main theorem of the article states: Let \(p\) be a prime not dividing the order of the Galois group, \(\Gamma= \text{Gal}(L/F)\) and assume that \(\text{Cl}(K/F)= 1\) for all normal extensions \(K/F\) with \(F\subseteq K\subset L\), but \(K\neq L\). Let \(\Phi\) denote the representation \(\Phi: \Gamma\to \operatorname{Aut}(C)\) induced by the action of \(\Gamma\) on \(C= \text{Cl}(L/F)/\text{Cl}(L/F)^p\); if \(C\neq 1\), then \(\Phi\) is faithful. If the degrees of all irreducible faithful \(F_p\)-characters \(\chi\) of \(T\) are divisible by \(f\) then the rank \(\text{Cl}_p(L)\equiv 0\pmod f\). If, in addition, \([F_p(\chi): F_p]= r\) for all these \(\chi\), then the rank \(\text{Cl}(L)_p\equiv 0\pmod{rf}\). Here \(F_p\) denotes the field of \(p\) elements and \(F_p(\chi)\) denotes the smallest extension field of \(F_p\) containing all values in the image of \(\chi\).
Applications of the results are given when \(\Gamma\) is the Klein 4-group, the quaternion and dihedral groups of order 8 and \(A_4\).


12F20 Transcendental field extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants
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[1] Burnside, W., On some properties of groups whose orders are powers of primes, Proc. London math. soc. (2), 11, 225-245, (1912) · JFM 43.0198.02
[2] Brown, E.; Parry, Ch., The 2-class group of certain biquadratic number fields, J. reine angew. math., 295, 61-71, (1977) · Zbl 0355.12007
[3] Brown, E.; Parry, Ch., The 2-class group of certain biquadratic number fields II, Pacific J. math., 78, 11-26, (1978) · Zbl 0405.12009
[4] Cohen, H.; Martinet, J., Étude heuristique des groupe de classes des corps de nombres, J. reine angew. math., 404, 39-76, (1990) · Zbl 0699.12016
[5] C. Cooper, Character tables of groups of order <100. Available at http://www.maths.mq.edu.au/ chris/chartab.htm
[6] Cornell, G.; Rosen, M., Group-theoretic constraints on the structure of the class group, J. number theory, 13, 1-11, (1981) · Zbl 0456.12005
[7] Gras, G., Sur LES ℓ-classes d’idéaux dans LES extensions cycliques relatives de degré premier ℓ, Ann. inst. Fourier, 23, 3, 1-48, (1973), ibid. 23 (4) 1-44 · Zbl 0276.12013
[8] Grün, O., Aufgabe 153; Lösungen von L. holzer und A. scholz, Jahresber. DMV, 45, 74-75, (1934), (kursiv)
[9] Halter-Koch, F., Einheiten und divisorenklassen in Galois’schen algebraischen zahlkörpern mit diedergruppe der ordnung 2ℓ für eine ungerade primzahl ℓ, Acta arith., 33, 353-364, (1977) · Zbl 0416.12003
[10] Huppert, B., Endliche gruppen I, Grundlehren der math. wiss., 134, (1967), Springer-Verlag · Zbl 0217.07201
[11] Inaba, E., Über die struktur der ℓ-klassengruppe zyklischer zahlkörper von primzahlgrad ℓ, J. fac. sci. Tokyo I, 4, 61-115, (1940) · JFM 66.0120.05
[12] Isaacs, I.M., Character theory of finite groups, (1976), Academic Press, 2nd edition, Dover, 1995 · Zbl 0337.20005
[13] Iwasawa, K., A note on ideal class groups, Nagoya math. J., 27, 239-247, (1966) · Zbl 0139.28104
[14] James, G.; Liebeck, M., Representations and characters of groups, (1993), Cambridge Univ. Press · Zbl 0792.20006
[15] Kihel, O., Extensions diédrales et courbes elliptiques, Acta arith., 102, 309-314, (2002) · Zbl 0996.11044
[16] Komatsu, T.; Nakano, S., On the Galois module structure of ideal class groups, Nagoya math. J., (2001) · Zbl 1045.11079
[17] Kondo, T., Some examples of unramified extensions over quadratic fields, Sci. rep. Tokyo woman’s Christian univ., 120-121, 1399-1410, (1997)
[18] Lecacheux, O., Constructions de polynômes génériques à groupe de Galois résoluble, Acta arith., 86, 207-216, (1998) · Zbl 0919.12002
[19] Lemmermeyer, F., On 2-class field towers of some imaginary quadratic number fields, Abh. math. sem. Hamburg, 67, 205-214, (1997) · Zbl 0919.11075
[20] Louboutin, S., Calcul des nombres de classes relatifs: application aux corps octiques quaternionique à multiplication complexe, C. R. acad. sci. Paris, 317, 643-646, (1993) · Zbl 0795.11059
[21] Martinais, D.; Schneps, L., Polynômes a groupe de Galois diédral, Semin. théor. nombres Bordeaux, 4, 141-153, (1992) · Zbl 0780.12006
[22] Roland, G.; Yui, N.; Zagier, D., A parametric family of quintic polynomials with Galois group D5, Number theory, 15, 137-142, (1982) · Zbl 0506.12011
[23] Smith, J., A remark on class numbers of number field extensions, Proc. amer. math. soc., 20, 388-390, (1969) · Zbl 0191.33403
[24] Tate, J.; Coates, J.; Helgason, S., LES conjectures de Stark sur LES fonctions L d’Artin en s=0, Progress in math., (1984), Birkhäuser
[25] Taussky, O., A remark on the class field tower, J. London math. soc., 12, 82-85, (1937) · JFM 63.0144.03
[26] Yamamura, K., Maximal unramified extensions of imaginary quadratic number fields of small conductors, J. théor. nombres Bordeaux, 9, 2, 405-448, (1997) · Zbl 0905.11048
[27] Yokoyama, A., Über die relativklassenzahl eines relativ-galoischen zahlkörpers von primzahlpotenzgrad, Tôhoku math. J., 18, 318-324, (1966) · Zbl 0148.27901
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