View source for Kaplansky field

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A [[Field|field]] $K$ equipped with a [[Valuation|valuation]] $v$ and satisfying Kaplansky's hypothesis A, as introduced in [[#References|[a2]]]. Let $p$ denote the characteristic of the residue field $Kv$ if it is a positive prime; otherwise, set $p=1$. Hypothesis A requires that:

i) the value group $vK$ is a [[P-divisible group|$p$-divisible group]];

ii) for every [[additive polynomial]] (cf. [[#References|[a5]]]) $f$ with coefficients in $Kv$ and every $a\in Kv$, the equation $f(X)=a$ has a solution in $Kv$. Requirement ii) implies that $Kv$ is a [[Perfect field|perfect field]]. Using [[Galois cohomology|Galois cohomology]], G. Whaples [[#References|[a6]]] showed that ii) means that the degree of every finite extension of $Kv$ is prime to $p$. An elementary proof of this was given by F. Delon in 1981.

I. Kaplansky considered immediate extensions (cf. [[Valuation|Valuation]]) of fields with valuations. He used pseudo-Cauchy sequences (also called Ostrowski nets), which were introduced by A. Ostrowski in 1935. These are analogues of Cauchy sequences (cf. [[Cauchy sequence|Cauchy sequence]]) for the metric induced by the valuation, but their limits need not be unique; for this reason, they are called pseudo-limits.

In [[#References|[a3]]], W. Krull proved that there always exist maximal immediate extensions (in order to apply the [[Zorn lemma]], Krull gave an upper bound for the cardinality of immediate extensions of a fixed field; an elegant deduction of this bound was later given by K.A.H. Gravett in [[#References|[a1]]]). Kaplansky showed that if $(K,v)$ satisfies hypothesis A, then its maximal immediate extensions are unique up to a valuation-preserving isomorphism over $K$. The same holds for the maximal immediate algebraic extensions, and this fact can also be shown via a Galois-theoretic interpretation of hypothesis A (cf. [[#References|[a4]]]). The former result follows from the latter by a theorem of Kaplansky (which has a certain analogue in the theory of [[real closed field]]s: If $K$ has no non-trivial immediate algebraic extensions, then the isomorphism type of an immediate extension $K(x)$ over $K$ is determined by the pseudo-Cauchy sequences in $K$ that have pseudo-limit $x$.

See also [[Model theory of valued fields]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> K.A.H. Gravett, "Note on a result of Krull" ''Proc. Cambridge Philos. Soc.'' , '''52''' (1956) pp. 379 {{MR|0075937}} {{ZBL|0073.02702}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Kaplansky, "Maximal fields with valuations I" ''Duke Math. J.'' , '''9''' (1942) pp. 303–321 {{MR|6161}} {{ZBL|0061.05506}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Krull, "Allgemeine Bewertungstheorie" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 160–196 {{MR|}} {{ZBL|0004.09802}} {{ZBL|58.0148.02}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> F.-V. Kuhlmann, M. Pank, P. Roquette, "Immediate and purely wild extensions of valued fields" ''Manuscr. Math.'' , '''55''' (1986) pp. 39–67 {{MR|0828410}} {{ZBL|0593.12018}} </TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Whaples, "Galois cohomology of additive polynomials and $n$-th power mappings of fields" ''Duke Math. J.'' , '''24''' (1957) pp. 143–150 {{MR|0092826}} {{ZBL|}} </TD></TR>
</table>