Maillet's determinant
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
- {{Citation | last1=Carlitz | first1=L. | author1-link=Leonard Carlitz | last2=Olson | first2=F. R. | title=Maillet's determinant | jstor=2032352 |mr=0069207 | year=1955 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=6 | issue=2 | pages=265–269 | doi=10.2307/2032352}}
- {{Citation | last1=Maillet | first1=E. | title=Question 4269 | url=https://books.google.com/books?id=0ABSAQAAIAAJ | year=1913 | journal=L'Intermédiaire des Mathématiciens | volume=xx | pages=218}}
- {{Citation | last1=Malo | first1=E. | title=Sur un certain déterminant d'ordre premier | url= http://babel.hathitrust.org/cgi/pt?id=pst.000052363436;view=1up;seq=189 | year=1914 | journal=L'Intermédiaire des Mathématiciens | volume=xxi | pages=173–176}}
- {{Citation | last1=Muir | first1=Thomas | title=Contributions To The History Of Determinants 1900–1920 | url=https://archive.org/details/contributionstot032405mbp | publisher=Blackie And Son Limited. | year=1930}}
- {{Citation | last1=Wang | first1=Kai | title=On Maillet determinant | journal=Journal of Number Theory |doi=10.1016/0022-314X(84)90064-7| publisher=Journal of Number Theory 18 | year=1984| volume=18 | issue=3 | pages=306–312 | doi-access=free }}
Category:Algebraic number theory
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