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Toward an arithmetic of polynomials. (English) Zbl 0759.13005

The authors investigate polynomials over commutative rings \(R\) having exactly two idempotents 0 and 1. The starting points are characterizations of \(s\)-separable polynomials \(f\in R[x]\) (i.e. there are \(u,v\in R[x]\) with \(uf+vf'=1)\) and the theorem that this condition on \(R\) is equivalent to the property that every monic \(s\)-separable polynomial can be factored uniquely into a product of monic irreducible polynomials. Then terms like finite-primes, real-primes of \(F\), unramified primes at \(f\in R[x]\) are defined. E.g. the first one denotes a pair \((A,P)\), where \(A\) is a finitely generated subring of \(R\) and \(P\) is a maximal proper ideal of \(A\); it specializes to the common primes for \(R=\mathbb{Q}\) in an obvious way. The significance of these terms is shown in number theoretical applications
partly known (e.g.: theorem 1.9: Let \(f=x^ 2-m\), \(m\) a squarefree integer, \(m\neq 0,1\), and \(q\in\mathbb{N}\), \(q\nmid m\). Then \((\mathbb{Z}[1/2,1/m]\), \(q\mathbb{Z}[1/2,1/m])\) is an unramified finite prime at \(f\) with inertial degree 1 if \(\left({m\over q}\right)=1\), and 2 otherwise) and
partly new as in the situation of generalized cyclotomic extensions \((f=\prod^ j_{i=1}(x^ n-p_ i)\) for certain primes \(p_ i\)).
Furthermore the category of monic \(s\)-separable polynomials is studied, in particular it is shown, that any irreducible such \(f\) has a normal closure if \(R\) is a field. Results concerning the cohomology of polynomials, the Amitsur complex and the cup product follow.
Reviewer: G.Kowol (Wien)

MSC:

13B25 Polynomials over commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11R09 Polynomials (irreducibility, etc.)
11R18 Cyclotomic extensions
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References:

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