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**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.

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)