A Proof Of Beiter-gallot-moree Conjecture

Cyclotomic fields are always central topics of algebraic number theory. In thisdissertation, we mainly study the ternary cyclotomic polynomials.LetΦ_n(x) be the nth cyclotomic polynomial.Φ_n(x) is said to be ternary if n is theproduct of three distinct odd primes, p < q < r. Let A(n) be the largest absolute valueof the coefficients ofΦ_n(x). In 1968, M. Beiter conjectured thatΦpqr≦(p+1)/2.M. Beiter proved her conjecture for p≦5 and also in case either q or r≡±1 (mod p).In 2009, Y. Gallot and P. Moree showed that the conjecture is false for every p≧11.For p = 7, it is still an open problem. Based on extensive numerical computations, theygave infinitely many counter-examples and proposed the Corrected Beiter conjecture:A(pqr)≦2/3 p. Moreover, this is the strongest corrected version of Beiter's conjectureof this form, since they also proved that there must exist odd primes p < q < r suchthat A(pqr) (2/3-ε)p for every sufficiently large prime p.In this paper, we first introduce a new method for computing the coefficients ofΦ_pqr(x). It is more effiective and explicit to estimate the upper bound of A(pqr). Thenwe obtain a sufficient condition of the Corrected Beiter conjecture and prove it whenp = 7. It is a definite answer to the aforementioned open problem. With the similarmethods, we finally prove the Corrected Beiter conjecture completely.When A(n) = 1,Φ_n(x) is said to be flat. N. Kaplan proved that if r≡±1(mod pq), then A(pqr) = 1. We give a new proof of this theorem and show a necessarycondition of A(pqr) = 1. We also get a more general result for the other cases of r(mod pq), which includes N. Kaplan's theorem as a special case. Moreover, we proveN. Kaplan's another theorem.G. Bachman proved that every integer shows up as a coefficient of a ternarycyclotomic polynomial. We give two new proofs using entirely different methods.