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Henselian function fields. (English) Zbl 0743.12008

Heidelberg: Univ. Heidelberg, Naturwiss.-Mathematische Fakultät, 270 p. (1989).
Valued fields are investigated traditionally from at least two points of view: valuation theory and model theory. Both these approaches are very closely connected and sometimes it is even impossible to separate them.
In the present thesis the author tries to develop both aspects of some parts of valued fields theory. The central role here play henselizations of valued function fields (called henselian function fields) and defectless fields, where a valued field \(F\) is called to be a defectless field if every finite extension \(E\) of \(F\) satisfies the basic equality \([E:F]=\sum e_ if_ i\). Although a henselization \(F^ h\) of a valued function field \(F\) is not a function field, it has many nice algebraic properties (for example, it satisfies the Hensel’s Lemma and it is an immediate extension).
In several chapters of this thesis the author investigates a structure theory of henselian function fields. He is mostly interested in henselian rational function fields \(F\) over the ground field \(K\), i.e. he finds certain assumptions under which \(F=K(x)^ h\). His principal results concern algebraically maximal perfect fields \(K\) of a positive characteristic. He proves (Theorem 8.11) that if \(F\) is an immediate henselian function field over \(K\) of a transcendence degree 1 over \(K\), then \(F\) is a henselian rational function field over \(K\). Moreover, he investigates in details the assumptions of this theorem. For example he shows that the condition “perfect” cannot be dropped.
The proof of the Theorem 8.11 is rather complicated and in a process of this proof the author gives a normal form for certain classes of immediate extensions, starting from a description of the structure of some purely wild algebraic extensions. These extensions are used to investigate some properties of tame fields of positive characteristic, where a field \(K\) is called tame if its ramification field \(K^ r\) is algebraically closed. The class of tame fields has a great importance for the results of this thesis since it comprises almost all assumptions on \(K\) under which a structure theory of henselian function fields may be developed. Namely it comprises algebraically closed valued fields, henselian fields of residue characteristic \(0\), algebraically maximal Kaplansky fields and algebraically maximal perfect fields of positive characteristic.
The other group of conditions under which a structure of henselian function fields may be determined concerns the situation where the henselian function field \(F/K\) has no transcendence defect, i.e. \[ \hbox{trdeg}(F/K)=\hbox{trdeg}(\overline F/\overline K)+rr(v(F)/v(K), \] where \(rr(v(F)/v(K))\) denotes the maximal number of rationally independent elements of \(v(F)/v(K)\). The principal result in this part is the Theorem 3.1, which states that if \(F/K\) is a henselian function field without transcendence defect and if \(K\) is a defectless field then \(F\) is a defectless field. Using this theorem the author proves that in this case \(F\) is a finite defectless extension of a henselian rational function field. This theorem allows then to generalize some notions and results for valued function fields. For example, the defect of \(F/K\) may be defined to be \[ d(F/K)=\hbox{sup}_{\mathcal T}d(F/K(\mathcal T)) \] where the supremum runs over all transcendence bases \(\mathcal T\) of \(F/K\) and \(d(F/K(\mathcal T))\) is a defect of an algebraic extension. The author then shows that if \(F/K\) has no transcendence defect, then \(d(F/K)\) equals \(d(F/K(\mathcal T))\) for every valuation transcendence bases \(\mathcal T\) of \(F/K\).
Furthermore, the author studies various generalizations of defects. For example, he introduces the notion of completion defect and defect quotient and he studies the behaviour of these functions (section 5). The principal results are connected with an inherence of various types of defectless from ground field \(K\) to a henselian function field without transcendence defect.
From the model theoretical point of view the author is interested in the so called AKE-classes, i.e. elementary classes of valued fields \((K,v)\) which satisfy Ax-Kochen-Ershov principle (AKE-fields): If \((L,v)/(K,v)\) is an extension of valued fields such that \(v(L) \prec_ \exists v(L)\) and \(\overline K \prec_ \exists \overline{L}\), then \((K,v) \prec_ \exists (L,v)\). It is well known that this class comprises for example henselian fields with residual characteristic zero, algebraically closed valued fields, \(p\)-adically closed fields etc. In addition to these results the author proves that every tame field of positive characteristic is an AKE-field (Theorem 9.5). Using some embedding lemma the author also proves that tame fields of positive characteristic are existentially closed in every immediate extension of transcendence degree 1 and by means of some example he shows that this does not remain true if “tame” is replaced by “algebraically complete”.
It is worthwhile to mention here a common valuation theoretical tool which the author uses instead of pseudo Cauchy sequences. This notion of approximation types is introduced in section 2 as a special filter and developed in section 12, where the author shows that every approximation type over a valued ground field is realized by an element in a valued field extension.
The present thesis represents a well written summary of approaches and results in different directions of valued fields theory and the new results are interesting and fruitful.

MSC:

12J10 Valued fields
12L05 Decidability and field theory
12L12 Model theory of fields
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