Higher product levels of domains

Abstract

The n-th product level of a skew-field D, ps n (D), is a generalization of the n-th level of a field F , s n (F ). An explicit bound for s 2m (F ) in terms of m and s 2 (F ) is known and it is also known that there is no such bound for ps 2m (D) when m is even. Our aim is to explicitly construct such a bound for odd m. More precisely, we construct a function f : N 3 → N, such that ps 2 k l (D) f (ps 2 k (D), k, l), for every integer k, every odd integer l and every skew-field D.We give an explicit bound for the n-th Pythagorean number of a simple extension Q(d)/Q in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n-th product level of an Ore domain is equal to the n-th product level of its skew field of fractions.

n-real valuations and the higher level version of the Krull–Baer theorem

“…Since O × v ⊂ P ∪ −P , ε ∈ −P . By Cimprič and Velušček [8,Corollary 4.3] it follows that ε ∈ D n ⊂ P and since is odd, also ε ∈ −P . Therefore ε ∈ P ∩ −P = {0}, a contradiction.…”

A Variant of Higher Product Levels of Integral Domains and *-Domains

“…Therefore, for a skew-field D and number n, we have besides s n (D) at least two other related invariants, ms n (D) and ps n (D), which are defined as the number of terms in the shortest representation of −1 as a sum of products (in the case of ms n ) or permuted products (in the case of ps n ) of n-th powers of elements from D. The invariant ps n (D) for n > 2 was introduced in [4] and studied later in [10] and [6]. The motivation for studying ps n comes from the theory of higher level orderings.…”