Characterization of extremal valued fields.

*(English)*Zbl 1271.12003This paper studies necessary and sufficient conditions for a valued field to be extremal, a notion with a somewhat involved history:

Yu. Ershov [Algebra Logika 43, No. 5, 582–588 (2004); translation in Algebra Logic 43, No. 5, 327–330 (2004; Zbl 1115.12002)] defined a valued field \((K,v)\) as extremal if for every polynomial \(f\in K[X_1,\dots,X_n]\) which has no zero in \(K\), the set of values \(v(f(a_1,\dots,a_n))\), where \(a_i\in K\) for all \(i\), has a maximal element in the value group, and he claimed that all discretely valued fields which are algebraically complete (i.e. Henselian and every finite extension is defectless) are extremal. A counterexample to this claim was given by Starchenko. In fact, a field which is extremal in this sense is always algebraically closed, as the authors show (Remark 3.2). They therefore replace Ershov’s definition by the following: A valued field \((K,v)\) is extremal if for every polynomial \(f\in K[X_1,\dots,X_n]\) which has no zero in \(K\), the set of values \(v(f(a_1,\dots,a_n))\), where \(v(a_i)\geq0\) for all \(i\), has a maximal element.

The authors prove that extremal in this modified sense indeed implies algebraically complete and is preserved under finite field extensions (Theorem 3.10). For residue characteristic zero, the authors then provide a complete characterization of extremal fields (Theorem 1.1): A valued field with residue field of characteristic zero is extremal if and only if the valuation is Henselian and either (1) the value group is a \(\mathbb{Z}\)-group (i.e. elementarily equivalent to the ordered group \(\mathbb{Z}\)) or (2) the value group is divisible and the residue field is large in the sense of F. Pop [Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)]. They show that, more generally, for a valued field \((K,v)\) without any assumption on the residue characteristic, if \(K\) is extremal, then it satisfies (1) or (2); and conversely, in the equicharacteristic case, if \(K\) is algebraically complete and satisfies (1) or (2), then it is extremal in case the residue field is perfect (Theorem 1.2), but need not be extremal otherwise (Theorem 1.3). In the last section they briefly discuss further topics like extremal formally \(p\)-adic fields.

There is a certain overlap of the results of the present paper with Yu. Ershov [Sib. Mat. Zh. 50, No. 6, 1280–1284 (2009); translation in Sib. Math. J. 50, No. 6, 1007–1010 (2009; Zbl 1224.12008)], in which Ershov introduces the notion of \(\ast\)-extremal fields, which coincides precisely with the modified definition of extremal in the present paper. Also Ershov proves that fields which are extremal in the original sense are algebraically closed, and that this new notion of extremal implies algebraic completeness and is preserved under finite extensions.

Yu. Ershov [Algebra Logika 43, No. 5, 582–588 (2004); translation in Algebra Logic 43, No. 5, 327–330 (2004; Zbl 1115.12002)] defined a valued field \((K,v)\) as extremal if for every polynomial \(f\in K[X_1,\dots,X_n]\) which has no zero in \(K\), the set of values \(v(f(a_1,\dots,a_n))\), where \(a_i\in K\) for all \(i\), has a maximal element in the value group, and he claimed that all discretely valued fields which are algebraically complete (i.e. Henselian and every finite extension is defectless) are extremal. A counterexample to this claim was given by Starchenko. In fact, a field which is extremal in this sense is always algebraically closed, as the authors show (Remark 3.2). They therefore replace Ershov’s definition by the following: A valued field \((K,v)\) is extremal if for every polynomial \(f\in K[X_1,\dots,X_n]\) which has no zero in \(K\), the set of values \(v(f(a_1,\dots,a_n))\), where \(v(a_i)\geq0\) for all \(i\), has a maximal element.

The authors prove that extremal in this modified sense indeed implies algebraically complete and is preserved under finite field extensions (Theorem 3.10). For residue characteristic zero, the authors then provide a complete characterization of extremal fields (Theorem 1.1): A valued field with residue field of characteristic zero is extremal if and only if the valuation is Henselian and either (1) the value group is a \(\mathbb{Z}\)-group (i.e. elementarily equivalent to the ordered group \(\mathbb{Z}\)) or (2) the value group is divisible and the residue field is large in the sense of F. Pop [Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)]. They show that, more generally, for a valued field \((K,v)\) without any assumption on the residue characteristic, if \(K\) is extremal, then it satisfies (1) or (2); and conversely, in the equicharacteristic case, if \(K\) is algebraically complete and satisfies (1) or (2), then it is extremal in case the residue field is perfect (Theorem 1.2), but need not be extremal otherwise (Theorem 1.3). In the last section they briefly discuss further topics like extremal formally \(p\)-adic fields.

There is a certain overlap of the results of the present paper with Yu. Ershov [Sib. Mat. Zh. 50, No. 6, 1280–1284 (2009); translation in Sib. Math. J. 50, No. 6, 1007–1010 (2009; Zbl 1224.12008)], in which Ershov introduces the notion of \(\ast\)-extremal fields, which coincides precisely with the modified definition of extremal in the present paper. Also Ershov proves that fields which are extremal in the original sense are algebraically closed, and that this new notion of extremal implies algebraic completeness and is preserved under finite extensions.

Reviewer: Arno Fehm (Konstanz)

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