Common Divisors of Values of Polynomials and Common Factors of Indices in a Number Field

Abstract

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.

A Survey on Fixed Divisors

“…The converse of the above theorem may not be true in general, however we have the following Theorem 6.4 (Ayad and Kihel [9]). Suppose that K is a Galois extension of Q.…”

“…McCluer [72] answered this question completely in 1971. Combining the notion of J(L|K), the above theorem and a classical result of Hensel (see [59] and [9]), Ayad and Kihel [9] gave one more interesting application of fixed divisors. Before proceeding we recall a few definitions.…”

“…Let S(L|K) be the set of elements a ∈ O L such that L = K(a) and f a (x) denote the minimal monic polynomial of a with coefficients in O K [x]. Building on these results, Ayad and Kihel [9] asked the following questions.…”

“…Connecting ρ(p) to the splitting of p, they conjectured Conjecture 6.5 (Ayad and Kihel [9]). If K is a Galois extension of degree n over Q and p | J(K|Q), then ρ(p) determines the splitting of p in K.…”

Indices in a Number Field

“…Moreover, we show that its values determine in some cases the splitting type of p in A. In [1], a function ρ p (K), similar to µ K (p), is defined.…”