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Solution of the class number two problem for cyclotomic fields. (English) Zbl 0288.12005
It is shown that the only cyclotomic fields of the form \(\mathbb Q(e^{2\pi i/m})\) which have class number two are \(\mathbb Q(e^{2\pi i/39})\) and \(\mathbb Q(e^{2\pi i/56})\). Methods are the same as used in solving the class number one problem [the author and H. L. Montgomery, J. Reine Angew. Math. 286/287, 248–256 (1976; Zbl 0335.12013)].
Reviewer: John Myron Masley

11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11R42 Zeta functions and \(L\)-functions of number fields
Full Text: DOI EuDML
[1] Baker, A.: Linear forms in the logarithms of algebraic numbers. Mathematika13, 204?216 (1966) · Zbl 0161.05201 · doi:10.1112/S0025579300003971
[2] Baker, A.: Imaginary quadratic fields with class number 2. Annals of Math.94, 139?152 (1971) · Zbl 0219.12008 · doi:10.2307/1970739
[3] Baker, A., Stark, H.M.: On a fundamental inequality in number theory. Annals of Math.94, 190?199 (1971) · Zbl 0219.12009 · doi:10.2307/1970742
[4] Bauer, H.: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. of Number Theory1, 161?162 (1969) · Zbl 0167.32301 · doi:10.1016/0022-314X(69)90034-1
[5] Carlitz, L.: A characterization of algebraic number fields with class number two. Proc. AMS11, 391?392 (1960) · Zbl 0202.33101
[6] Hasse, H.: Über die Klassenzahl aberscher Zahlkörper. Berlin: Academic Verlag 1952 · Zbl 0046.26003
[7] Iwasawa, K.: A note on class numbers of algebraic number fields. Abh. Math. Sem. Univ. Hamburg20, 257?258 (1956) · Zbl 0074.03002
[8] Iwasawa, K.: A note on ideal class groups. Nagoya Math. J.27, 239?247 (1966) · Zbl 0139.28104
[9] Masley, J.: On the class number of cyclotomic fields. Dissertation, Princeton Univ. 1972
[10] Masley, J., Montgomery, H.L.: Unique factorization in cyclotomic fields. To appear · Zbl 0335.12013
[11] Metsankyla, T.: On prime factors of the relative class numbers of cyclotomic fields. Ann. Univ. Turku. Ser A I, 149 (1971)
[12] Metsankyla, T.: On the growth of the first factor of the cyclotomic class number. Ann. Univ. Turku. Ser A I, 155 (1972)
[13] Montgomery, H.L., Weinberger, P.: Notes on small class numbers, to appear · Zbl 0285.12004
[14] Schrutka v. Rechtenstamm, G.: Tabelle der (relativ.) Klassenzahlen von Kreiskörpern, Abh. Deutsche Akad. Wiss. Berlin, 1964, Math. Nat. K1. Nr. 2 · Zbl 0199.09803
[15] Stark, H. M.: A complete determination of the complex quadratic fields of class-number one. Mich. Math. J.14, 1?27 (1967) · Zbl 0148.27802 · doi:10.1307/mmj/1028999653
[16] Stark, H. M.: A transcendence theorem for class-number problems. Annals of Math.94, 153?173 (1971) · Zbl 0229.12010 · doi:10.2307/1970740
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