Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcm θ∈Âi(θ), where i(θ) = gcd x∈ℤFθ(x) and Fθ(x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρK(p) defined as the number of [Formula: see text] such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρK(p) are stated.
Functions counting the number of subsets of ¹1; 2; : : : ; nº having particular properties are defined by Nathanson. Here, generalizations in two directions are given.
A nonempty subset A ¹1; 2; : : : ; nº is relatively prime if gcd.A/ D 1. Let f .n/ denote the number of relatively prime subsets of ¹1; 2; : : : ; nº. The sequence given by the values of f .n/ is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. In this article we show that f .n/ is never a square if n 2. Moreover, we show that reducing the terms of this sequence modulo any prime l ¤ 3 leads to a sequence which is not periodic modulo l.
Let F q be the finite field of characteristic p containing q = p r elements and f (x) = ax n + x m , a binomial with coefficients in this field. If some conditions on the greatest common divisor of n − m and q − 1 are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if f (x) = ax n + x m permutes F p , where n > m > 0 and a ∈ F * p , then p − 1 ≤ (d − 1)d, where d = gcd(n − m, p − 1), and that this bound of p, in terms of d only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of F q from a permutation binomial over F q .
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