Some Problems Of Cyclotomic Field

In this paper, we mainly study the coefficients of the cyclotomic polynomials, the monotonicity of the cyclotomic polynomials, and the determinantal formula for the special values of the Dedekind zeta function of the cyclotomic field. The main content as follows:1. We introduce some concepts and properties of cyclotomic polynomial and the Dedekind zeta function and so on.2. We discuss the coefficients and the monotonicity of the cyclotomic polynomials. We get the results below: 1°. For any integer l≥0, we havewhereα_2^ln(k) is the kth coefficient of the 2^lnth cyclotomic polynomial. 2°. Let n≥3 be any integer, then we haveΦ_n(x) is strictly increasing for x > 1 and strictly decreasing for x <—1.3. We study the determinantal formula for the special values of the Dedekind zeta function of the cyclotomic field. We prove the below theorem: If K = is P1P2…p_tth cyclotomic field and K⁺ is its maximal real subfield, where p1, p2,…, p_t are distinct odd primes, then(1) For any positive odd integer n, we have(2) For any positive even integer n, we have where...