Permutations And Determinants Related To Quadratic Residues

In first part of this thesis,we study signatures of permutations related to quadratic residues or primitive roots modulo an odd prime.Given an odd prime,let a1<?a2<…<a?p-1?/2 be all the quadratic residues moduloamong 1,2,...,-1.Consider the following sequences:A0:a1,a2,…,a?p-1?/2,A1:{12}p,{22}p,...,{?p-1/2?2}p,A2:{22}p,{42}P,...,{?p-1?2}p,A3:{12}p,{32}P,...,{?p-2?2}p,A4+{1?1/p?}p;{2?2/p?}p,...,{p-1/2??p-1?/2/p?}p,where?????denotes the Legendre symbol and{a}n denotes the least nonnegative residue of a modulo n.When Ai is a permutation of Aj,we denote this permutation by ?i,j.We determine the signatures of all these permutations.For instance,when p?3?mod 4? we show that???,where h?-p?denotes the class number of the imaginary quadratic field Q?????.We also study permutations on residues modulo prime powers.Letbe an odd prime,and letbe any positive integer.Let b1<b2<...<bn be the integers among 1,2,...,pr-1 which are coprime to pr.Then n=??pr?=pr-1?p-1? where??·? is Euler's totient function?,and {b1,b2,...bn} is a reduced system of residues modulo pr.Let Rpr={1?g<pr:g is a primitive root modulo pr.For any g ? Rpr,we define a permutation ?g on {b1,b2,...bn} by ?g:bi?{gi}pr?i=1,2,...,n?.In the case r=1,the signature of ?g was proposed by S.Kohl and solved by F.Ladisch and F.Petrov jointly.We obtain the following general result.?i?If?1?mod 4?,then|{g ? Rpr:sgn??g?=1}|=|{g ?Rpr:sgn??g?=-1}|.?ii?If?3?mod 4?,then for each g ?Rpr we have sgn??g?=?-1?h?-p?-1/2.Suppose that g ?Rp is a primitive root modulo an odd prime.Inspired by Kohl's problem and the Gauss'Lemma in the theory of quadratic residues,for any odd primewe consider a permutation of h_p={1,2,...,?p-1?/2}.Let g ?Rp be a primitive root modulo,and define a bijection from H-pto itself as follows:???We determine the signature of₂).?i?If?1?mod 8?,then for each g ?Rp we have sgn??g?=?-1?1/4h?-4p?·sgn??g?where h?-4p?denotes the class number of the imaginary quadratic field Q???.?ii?If?5?mod 8?,then for each g?Rp we have sgn??g?=?-1?1/4?h?-4p?+2??iii?If p is of the form 18?2n-1?²+1?where n is a positive integer?,then sgn??g?=?-1?n?iv?If?3?mod 4?,p>3 and p?18?2n-1?²+1 for all n=1,2,3,...,then|{g?Rp:sgn??g?=1}|=|{g?Rp:sgn??g?=-1}|.?0.8?The above results have been published or accepted.The remaining contents are about the evaluations of determinants involving Legen-dre symbol entries.Z.-W.Sun studied many determinants involving Legendre symbol entries and also posed several conjectures.We confirm two of them.For another con-jecture,we obtain a weaker result.Our main tools are Matrix-Determinant Lemma,Gauss sums and Lagrange's interpolation formula.For an odd prime,Sun defined M_P,M⁺_P and N⁺_P as follows.M_P is the matrix obtaining from [?i-j/p?]0?i,j??p-1?/2 vi-a replacing all the entries in the first row by 1,M⁺_P is the matrix obtaining from[?i+j/p?]0<i,j??p-1?/2 via replacing all the entries in the first row by 1,and N⁺_P is the matrix obtaining from[?i+j/p?]1<i,j??p-1?/2 via replacing all the entries in the first row by1.We prove the following results.?i?????ii?????iii????.This thesis consists of five chapters.In Chapter 1,we introduce the background of permutation and determinant problems and present our main results as well as sketches of their proofs.Chapters 2–5 are devoted to our detailed proofs of the main results in this thesis.