mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Maillet's determinant ''D''_''p'' is the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

introduced by whose entries are ''R''(''s/r'') for ''s'',''r'' = 1, 2, ..., (''p'' – 1)/2 ∈ Z/''p''Z for an odd prime ''p'', where and ''R''(''a'') is the least positive residue of ''a'' mod ''p'' . calculated the determinant ''D''_''p'' 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. showed that this conjecture is incorrect; the determinant in general is given by ''D''_''p'' = (–''p'')^{(''p'' – 3)/2}''h''⁻, where ''h''⁻ is the first factor of the class number of the

cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...

generated by ''p''th 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

* * * * * Algebraic number theory Determinants {{numtheory-stub