| 词条 | Maillet's determinant |
| 释义 |
In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by {{harvtxt|Maillet|1913}} whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p {{harv|Muir|1930|loc=pages 340–342}}. {{harvtxt|Malo|1914}} calculated the determinant Dp for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (–p)(p – 3)/2, and conjectured that it is given by this formula in general. {{harvtxt|Carlitz|Olson|1955}} showed that this conjecture is incorrect; the determinant in general is given by Dp = (–p)(p – 3)/2h−, where h− is the first factor of the class number of the cyclotomic field generated by pth roots of 1, which happens to be 1 for p less than 23. In particular this verifies Maillet's conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it. Their results have been extended to all non-prime odd numbers by K. Wang(1982). References
2 : Algebraic number theory|Determinants |
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