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.


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


Zbl 0603.12002
Full Text: DOI