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Irreducible polynomials which are locally reducible everywhere. (English) Zbl 1134.11040
Let $$K$$ be a global field, $$P(K)$$ the set of nontrivial places of $$K$$, $$O _ K$$ a Dedekind subring of $$K$$ with a fraction field $$K$$, $$M(K)$$ the set of maximal ideals of $$O _ K$$, and $$f(X) \in O _ K[X]$$ an irreducible polynomial over $$k$$ in one indeterminate (for example, assume that $$O _ K$$ is the maximal order of $$K$$, provided that char$$(K) = 0$$).
It is known (and easily obtained from Chebotarev’s Density Theorem as well as from Bauer’s theorem) that if the degree $$n$$ of $$f$$ is a prime number, then $$f$$ remains irreducible over infinitely many $$p \in M(K)$$. When $$n$$ is composite and $$p$$ runs across the elements of $$M(K)$$ not containing the discriminant of $$f$$, one deduces from Chebotarev’s theorem and Galois theory that $$f$$ is reducible modulo $$p$$ (or equivalently, by Hensel’s lemma, over the completion $$K _ p$$ of $$K$$ with respect to the $$p$$-adic topology) if and only if the Galois group $$G _ f$$ of $$f$$ over $$K$$ does not contain an element of order $$n$$.
Generally, the irreducibility of $$f$$ over $$K _ p\colon \;p \in P(K)$$, implies that $$f$$ is irreducible over every $$p \in M(K)$$ (but not the converse). Also, the fulfillment of the noted condition on $$G _ f$$ does not guarantee irreducibility of $$f$$ modulo every $$p \in M(K)$$.
On this basis, the paper under review proves, for each composite positive integer $$m$$, the existence of a polynomial $$g _ m \in O_ K [X]$$ of degree $$m$$, which is irreducible over $$k$$ but is reducible over $$K _ p$$, for all $$p \in P(K)$$; this generalizes a result obtained by R. Brandl (see [Am. Math. Mon. 93, 286–288 (1986; Zbl 0603.12002)]).
For the proof, the authors consider separately the cases where char$$(k) = 0$$ and char$$(k) \neq 0$$. Their argument also depends on whether or not $$m$$ is square-free.

##### MSC:
 11R09 Polynomials (irreducibility, etc.) 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11S25 Galois cohomology 12F05 Algebraic field extensions 12G05 Galois cohomology
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