The Arithmetic Of Cyclotomic Polynomials And Sums Of Squares In Number Fields

This thesis is mainly focused on the coefficients of cyclotomic polynomials and sums of three integral squares in the ring of algebraic integers in biquadratic fields.1. Let n be a positive integer and denote the n-th cyclotomic polynomial, where (?) is the Euler totient function. Let the height of Φn(x), written as A(n), be the largest absolute value of the coefficients of Φn(x). We say that Φn(x) is flat if A(n)= 1, and ternary if n is a product of three distinct odd primes, respectively. Our main results are as follows:(1.1) We give the explicit value for the coefficient of xr in ternary cyclotomic polynomial Φpqr (x), where p< q< r are odd primes.(1.2) Let p< q< r be odd primes such that zr≡ ±1 (mod pq), where z is a positive integer. We classify all flat ternary cyclotomic polynomials in the cases z= 3,4,5.(1.3) Let p< q< r be odd primes satisfying p≡ 1 (mod 3), q≡ 2p+2 (mod 3p) and r≡±3 (mod pq). We show that A(pqr)= 3. This provides an infinite family of ternary cyclotomic polynomials with height exactly 3, without fixing P.(1.4) Let p< q< r be odd primes with q(?)1 (mod p) and r≡- 2 (mod pq). We construct an explicit k such that a(pqr, k) = -2.(1.5) Let g(f) denote the maximum of the differences (gaps) between two con-secutive exponents occurring in a polynomial f. We give a new proof of g (Φpq)= p - 1, and show that the number of maximum gaps in Φpq(x) is given by 2[q/p], where p< q are odd primes.2. Let K be an algebraic number field and OK the ring of integers in K. Let SK be the set of all elements α∈OK which are sums of squares in OK and s(OK) the minimal number of squares necessary to represent -1 in OK.Let g(SK) be the smallest positive integer t such that every element in SK is a sum of t squares in OK. For K=Q((?)-m,(?)-n),where m≡n≡3(mod 4)are two distinct positive square-free integers,we obtain the following results:(2.1)SK=OK.(2.2)Ifs(OK)=2,then g(OK)=3.(2.3)In the case where m has no more than three distinct prime factors,we give some sufficient conditions such that g(OK)=3.