Abstract:
Let \lcub;K m } m ≥ 4 be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P m (x) =x 3−mx 2−(m+1)x− 1, m≥ 4. We determine all these K m 's with class numbers h m ≤ 3: there are 14 such K m 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents e m of the ideal class groups of these K m go to infinity with m and we determine all these K m 's with ideal class groups of exponents e m ≤ 3: there are 6 suchK m with ideal class groups of exponent 2, and 6 such K m with ideal class groups of exponent 3.
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Received: 16 November 2000 / Revised version: 16 May 2001
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Louboutin, S. Class number and class group problems for some non-normal totally real cubic number fields. manuscripta math. 106, 411–427 (2001). https://doi.org/10.1007/s229-001-8025-7
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DOI: https://doi.org/10.1007/s229-001-8025-7