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| On these pages, we present most recent results of our joint research, directly from the lab. |
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58 |
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E-mail addresses:
Karim.Belabas@math.u-psud.fr aderhem@yahoo.fr danielmayer@algebra.at |
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| Quadratic fields with cyclic 2-class group (2002/02/17) |
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Aïssa (02/02/15):
By the way,
what about the 2-class number of a
quadratic field whose conductor is divisible by 2 primes? Is it bounded? Do you have tables giving such data? |
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Dan (02/02/15):
I will try to answer your question on quadratic 2-class numbers.
In the first moment, I would say, this is Gauss' classical theory of genera, but I have to think a little bit about it. |
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Dan (02/02/17):
My first guess, that Gauss' classical theory of genera provides an answer for your question,
was not correct. The problem you posed was to get a statement about the 2-class number of a very specific kind of quadratic fields: those with exactly 2 prime divisors q_1, q_2 of their conductor f. Gauss' result was, that a quadratic field, whose conductor is divisible exactly by t prime factors, has 2-class rank rho_2 = t - 1. Thus, your desired kind of quadratic fields has rho_2 = 1, i. e., non-trivial cyclic 2-class group. 1. First, I studied the tables of complex and real quadratic fields in the appendix of Z. I. Borevich and I. R. Shafarevich, "Number Theory", Pure and Applied Math. 20, Academic Press, 1966 a) among the complex fields with squarefree radicands 0 > R > -500, I found the minimal radicands with the given class number h (and with discriminant d): -------------------------------------- R | q_1 | q_2 | d | h -------|-------|-------|--------|----- -5 | 2 | 5 | -20 | 2 -14 | 2 | 7 | -56 | 4 -41 | 2 | 41 | -164 | 8 -146 | 2 | 73 | -584 | 16 -446 | 2 | 223 | -1784 | 32 -------------------------------------------- The minimal discriminants, however, are: -------------------------------------- d | q_1 | q_2 | R | h -------|-------|-------|--------|----- -15 | 3 | 5 | -15 | 2 -39 | 3 | 13 | -39 | 4 -95 | 5 | 19 | -95 | 8 -407 | 11 | 37 | -407 | 16 -791 | 7 | 113 | -791 | 32 ( taken from 2. ) -------------------------------------------- b) among the real fields with squarefree radicands 0 < R < 500, I found the minimal radicands with the given class number h (and with discriminant d): -------------------------------------- R | q_1 | q_2 | d | h -------|-------|-------|--------|----- 10 | 2 | 5 | 40 | 2 82 | 2 | 41 | 328 | 4 226 | 2 | 113 | 904 | 8 2305 | 5 | 461 | 2305 | 16 ( taken from 2. ) -------------------------------------------- 2. Second, I browsed through the tables of complex and real quadratic fields in the appendix of R. A. Mollin, "Quadratics", CRC Press, 1995, where I found the two supplements above. Further I saw that there does not exist a) a squarefree radicand 0 > R > -2000 with h = 64 b) a squarefree radicand 0 < R < 10000 with h = 32 3. Finally, it came to my mind that I myself had calculated class groups of complex quadratic fields in autumn 1990 at Winnipeg City for 0 > d > -10^6. In the corresponding databases, I found: -------------------------------------------- d | q_1 | q_2 | R | h ---------|-------|-------|----------|------- -2519 | 11 | 229 | -2519 | 64 -13295 | 5 | 2659 | -13295 | 128 -43919 | 37 | 1187 | -43919 | 256 -132599 | 97 | 1367 | -132599 | 512 -328319 | 397 | 827 | -328319 | 1024 -------------------------------------------- But no example with h = 2048 downto -10^6. However, with an extension of the table, I could imagine that we also find h_2 = 2^{11}, and so on. Probably, the 2-class number of your desired kind of fields is unbounded. |
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