Abstract
Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer''s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1].
Highlights
Let K be a number field of degree n over Q and let A be its ring of integers
On the Indices in Number Fields and Their Computation for Small Degrees. 29 and K2 = Q(θ) : θ6 + θ5 + θ4 + θ3 + θ2 + θ + 1 = 0, are number fields of degree 6, we prove that the prime 2 has the same splitting type P1P2 in Ki, i = 1, 2 and v2(i(K1)) = 4, v2(i(K2)) = 3
We can suppose that K = Q(θ), where θ is a root of an irreducible polynomial of the type f (x) = x3 − ax + b, a, b ∈ Z
Summary
Let K be a number field of degree n over Q and let A be its ring of integers. Denote by A = {θ ∈ A such that K = Q(θ)} the set of primitive elements of A. 29 and K2 = Q(θ) : θ6 + θ5 + θ4 + θ3 + θ2 + θ + 1 = 0, are number fields of degree 6, we prove that the prime 2 has the same splitting type P1P2 in Ki, i = 1, 2 and v2(i(K1)) = 4, v2(i(K2)) = 3 (see example 1). P has the same splitting type in K1 and K2 They make the following conjecture in [1] : let K be a Galois number field and p a prime number such that p | i(K).
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