On the Coefficients of the Minimal Polynomials of Gaussian Periods
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چکیده
Let / be a prime number and m a divisor of /— 1. Then the Gauss period w = f + Çx + if +•■• + {/* , where ( = e2n'll and A is a primitive wth root of unity modulo /, generates a subfield K of Q(f ) of degree (/ 1)1 m . In this paper we study the reciprocal minimal polynomial Fi,m(X) = NK/q(1 ojX) of (w over Q. It will be shown that for fixed m and every N we have FLm(X) = (Bm(X)'/(l mX))l/m (mod**) for all but finitely many "exceptional primes" / (depending on m and N), where Bm(X) e Z[[X]] is a power series depending only on m . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite / is also discussed. 1. Statement of results Let / be an odd prime number1 and I X = m-d a decomposition of / 1 into positive factors. Then there is a unique cyclic extension Kd/Q of degree d ramified only at /. It is contained in the cyclotomic field Q(Ç) (Ç = primitive /th root of unity) and is generated over Q by the Gaussian period °> = TrQ«)/Jc, = C + CA + C"2 + • • • + Cx""', where X £ (Z//Z)x is a primitive mth root of unity modulo /. The minimal polynomial of œ, fi,m{X)=l\(x (c + ̂ + c^ + .-. + i^-1')), where ¿% denotes a set of coset representatives for (Z//Z)x modulo (X), gives an explicit irreducible polynomial of degree d with cyclic Galois group and discriminant a power of /. We include / and m rather than / and d into the notation because we will be studying the coefficients of these polynomials for m fixed and / varying. Specifically, we will show that for m and n fixed the «th coefficient "from the end" of fi,m(X) is a polynomial in / for all but finitely many "exceptional" primes /, and we will describe the computation of this polynomial and of the set of exceptional primes. The statement about the nth coefficient being a polynomial in / for / large, and some of our other Received by the editor August 19, 1991 and, in revised form, January 3, 1992. 1991 Mathematics Subject Classification. Primary 11L05, 11T22; Secondary 11Y40. 'The case of composite / will be considered briefly in §5. ©1993 American Mathematical Society 0025-5718/93 $1.00 + $.25 per page
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تاریخ انتشار 2010