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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

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions 11R42 Zeta functions and $$L$$-functions of number fields
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##### References:
 [1] Baker, A.: Linear forms in the logarithms of algebraic numbers. Mathematika13, 204?216 (1966) · Zbl 0161.05201 [2] Baker, A.: Imaginary quadratic fields with class number 2. Annals of Math.94, 139?152 (1971) · Zbl 0219.12008 [3] Baker, A., Stark, H.M.: On a fundamental inequality in number theory. Annals of Math.94, 190?199 (1971) · Zbl 0219.12009 [4] Bauer, H.: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. of Number Theory1, 161?162 (1969) · Zbl 0167.32301 [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 [16] Stark, H. M.: A transcendence theorem for class-number problems. Annals of Math.94, 153?173 (1971) · Zbl 0229.12010
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