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A [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100201.png" /> equipped with a [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100202.png" /> and satisfying Kaplansky's hypothesis A, as introduced in [[#References|[a2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100203.png" /> denote the characteristic of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100204.png" /> if it is a positive prime; otherwise, set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100205.png" />. Hypothesis A requires that:
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A [[Field|field]] $K$ equipped with a [[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100206.png" /> is a [[P-divisible group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100207.png" />-divisible group]];
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i) the value group $vK$ is a [[P-divisible group|$p$-divisible group]];
  
ii) for every additive polynomial (cf. [[#References|[a5]]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100208.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k1100209.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002010.png" />, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002011.png" /> has a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002012.png" />. Requirement ii) implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002014.png" /> is prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002015.png" />. An elementary proof of this was given by F. Delon in 1981.
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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]]. 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.
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I. Kaplansky considered [[immediate extension]]s of fields with [[valuation]]s. 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]]) 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002016.png" /> satisfies hypothesis A, then its maximal immediate extensions are unique up to a valuation-preserving isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002017.png" />. 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 fields, cf. [[Real closed field|Real closed field]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002018.png" /> has no non-trivial immediate algebraic extensions, then the isomorphism type of an immediate extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002020.png" /> is determined by the pseudo-Cauchy sequences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002021.png" /> that have pseudo-limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002022.png" />.
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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|Model theory of valued fields]].
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See also [[Model theory of valued fields]].
  
 
====References====
 
====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</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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Krull,   "Allgemeine Bewertungstheorie" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 160–196</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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Lang,   "Algebra" , Addison-Wesley (1974)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Whaples,   "Galois cohomology of additive polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110020/k11002023.png" />-th power mappings of fields" ''Duke Math. J.'' , '''24''' (1957) pp. 143–150</TD></TR></table>
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<table>
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<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>
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<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>
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<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>
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<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>
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<TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR>
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<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>
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</table>

Latest revision as of 19:54, 1 January 2015

A field $K$ equipped with a valuation $v$ and satisfying Kaplansky's hypothesis A, as introduced in [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;

ii) for every additive polynomial (cf. [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. Using Galois cohomology, G. Whaples [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 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) for the metric induced by the valuation, but their limits need not be unique; for this reason, they are called pseudo-limits.

In [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 [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. [a4]). The former result follows from the latter by a theorem of Kaplansky (which has a certain analogue in the theory of real closed fields: 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

[a1] K.A.H. Gravett, "Note on a result of Krull" Proc. Cambridge Philos. Soc. , 52 (1956) pp. 379 MR0075937 Zbl 0073.02702
[a2] I. Kaplansky, "Maximal fields with valuations I" Duke Math. J. , 9 (1942) pp. 303–321 MR6161 Zbl 0061.05506
[a3] W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1932) pp. 160–196 Zbl 0004.09802 Zbl 58.0148.02
[a4] F.-V. Kuhlmann, M. Pank, P. Roquette, "Immediate and purely wild extensions of valued fields" Manuscr. Math. , 55 (1986) pp. 39–67 MR0828410 Zbl 0593.12018
[a5] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[a6] G. Whaples, "Galois cohomology of additive polynomials and $n$-th power mappings of fields" Duke Math. J. , 24 (1957) pp. 143–150 MR0092826
How to Cite This Entry:
Kaplansky field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kaplansky_field&oldid=14681
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article