## Irreducibility over valued fields in the presence of a secondary valuation.(English)Zbl 1062.11075

A classical fact concerning irreducible polynomials over a field $$K$$ which is complete with respect to a non-Archimedean absolute value $$|\cdot|$$ is that if $$f(X)= X^d+ a_1 X^{d-1}+\cdots+ a_d\in K[X]$$ is irreducible over $$K$$, then there exists an $$\varepsilon> 0$$ such that any polynomial $$g(X)= X^d+ b_1X^{d-1}+\cdots+ b_d\in K[X]$$ with $$|b_j- a_j|< \varepsilon$$ for any $$j$$ is also irreducible over $$K$$.
In this paper, the author presents an irreducibility result as above, which is applicable to some valued fields $$K$$ that are not necessarily complete with respect to the given absolute value. Here the lack of completeness of the field $$K$$ is compensated by the existence of a secondary valuation on $$K$$, which satisfies a certain property in connection with the given primary valuation. More precisely, the author proves the following:
Theorem. Let $$K$$ be a field of characteristic zero, equipped with two non-Archimedean valuations $$v_1$$ and $$v_2$$, and let $$A$$ be a subring of $$K$$ with field of fractions $$K$$, which is integrally closed in $$K$$. Denote by $$\Gamma_1$$ and $$\Gamma_2$$ the value groups of $$v_1$$ and $$v_2$$, respectively. Assume also that no element from $$\overline A\setminus A$$ can be approximated with respect to $$\overline v_1$$ by a sequence of elements from $$A$$ which is bounded with respect to $$v_2$$, where $$\overline A$$ is the integral closure of a fixed algebraic closure $$\overline K$$ of $$K$$ and $$\overline v_i$$ is the extension of $$v_i$$ to $$\overline K$$ for each $$i$$. Then for any polynomial $$f(X)= X^d+ a_1 X^{d-1}+\cdots+ a_d\in A[X]$$ which is irreducible over $$K$$ and for any $$\gamma_2\in\Gamma_2$$, there exists $$\gamma_1\in\Gamma_1$$ such that for any $$b_1,\dots, b_d\in A$$ for which $$v_1(b_i- a_i)\geq \gamma_1$$ and $$v_2(b_i)\geq \gamma_2$$ for each $$i$$, the polynomial $$g(X)= X^d+ b_1X^{d-1}+\cdots+ b_d$$ is irreducible over $$K$$.
The author also applies the above theorem to an algebraic function field of one variable over a $$p$$-adic field by taking two non-Archimedean valuations on it.
Reviewer: Kaori Ota (Tokyo)

### MSC:

 11S05 Polynomials 11C08 Polynomials in number theory 12J10 Valued fields