| Popis: |
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 2 m ( F ) in terms of m and s 2 ( F ) is known and it is also known that there is no such bound for ps 2 m ( 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. |
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