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''of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844701.png" />''
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{{TEX|done}}
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{{MSC|12Fxx}}
  
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844702.png" /> such that for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844703.png" /> the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844705.png" /> are linearly disjoint over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844706.png" /> (see [[Linearly-disjoint extensions|Linearly-disjoint extensions]]). An extension that is not separable is called inseparable. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844707.png" /> is the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844708.png" />. In characteristic 0 all extensions are separable.
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A separable extension of a field $k$''
 +
is an extension $K/k$ such that for some natural number $n$ the fields $K$
 +
and $k^{p^{-n}}$ are linearly disjoint over $k$ (see
 +
[[Linearly-disjoint extensions|Linearly-disjoint extensions]]). An
 +
extension that is not separable is called inseparable. Here $p$ is the
 +
characteristic of $k$. In characteristic 0 all extensions are
 +
separable.
  
In what follows only algebraic extensions will be considered (for transcendental separable extensions see [[Transcendental extension|Transcendental extension]]). A finite extension is separable if and only if the [[Trace|trace]] mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s0844709.png" /> is a non-zero function. An algebraic extension is separable if any finite subextension is separable.
+
In what follows only algebraic extensions will be considered (for
 +
transcendental separable extensions see
 +
[[Transcendental extension|Transcendental extension]]). A finite
 +
extension is separable if and only if the
 +
[[Trace|trace]] mapping ${\mathrm Tr} : K\to k$ is a non-zero function. An algebraic
 +
extension is separable if any finite subextension is separable.
  
The separable extensions form a distinguished class of extensions, that is, in a [[Tower of fields|tower of fields]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447010.png" /> the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447011.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447013.png" /> are separable; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447015.png" /> are separable extensions, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447016.png" />; for a separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447017.png" /> and an arbitrary extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447018.png" /> the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447019.png" /> is again separable. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447020.png" /> is separable if and only if it admits an imbedding in a [[Galois extension|Galois extension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447021.png" />. In this case, the number of different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447022.png" />-isomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447023.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447024.png" /> is the same as the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447025.png" /> for a finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447026.png" />. Any finite separable extension is simple.
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The separable extensions form a distinguished class of extensions,
 +
that is, in a
 +
[[Tower of fields|tower of fields]] $L\supset K\supset k$ the extension $L/k$ is separable
 +
if and only if $L/K$ and $K/k$ are separable; if $K_1/k$ and $K_2/k$ are separable
 +
extensions, then so is $K_1K_2/k$; for a separable extension $K/k$ and an
 +
arbitrary extension $L/k$ the extension $KL/L$ is again separable. An
 +
extension $K/k$ is separable if and only if it admits an imbedding in a
 +
[[Galois extension|Galois extension]] $L/k$. In this case, the number of
 +
different $k$-isomorphisms of $K$ into $L$ is the same as the degree
 +
$[K:k]$ for a finite extension $K/k$. Any finite separable extension is
 +
simple.
  
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447027.png" /> is called separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447028.png" /> if none of its irreducible factors has a multiple root in an algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447029.png" />. An algebraic element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447030.png" /> is called separable (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447031.png" />) if it is a root of a polynomial that is separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447032.png" />. Otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447033.png" /> is called inseparable. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447034.png" /> is called purely inseparable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447035.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447036.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447037.png" />. An irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447038.png" /> is inseparable if and only if its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447039.png" /> is identically zero (this is possible only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447040.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447042.png" />). An arbitrary irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447043.png" /> can be uniquely represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447045.png" /> is a separable polynomial. The degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447046.png" /> and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447047.png" /> are called, respectively, the reduced degree and the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447048.png" />.
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A polynomial $f\in k[x]$ is called separable over $k$ if none of its
 +
irreducible factors has a multiple root in an algebraic closure of
 +
$k$. An algebraic element $\def\a{\alpha}$ is called separable (over $k$) if it is
 +
a root of a polynomial that is separable over $k$. Otherwise $\a$ is
 +
called inseparable. An element $\a$ is called purely inseparable over
 +
$k$ if $\a^{p^n}\in k$ for some $n$. An irreducible polynomial $f(x)$ is inseparable
 +
if and only if its derivative $f'(x)$ is identically zero (this is
 +
possible only for $k$ of characteristic $p>0$ and $f(x)=f_1(x^p)$). An arbitrary
 +
irreducible polynomial $f(x)$ can be uniquely represented in the form
 +
$f(x)=g(x^{p^e})$, where $g(x)$ is a separable polynomial. The degree of $g(x)$ and the
 +
number $e$ are called, respectively, the reduced degree and the index
 +
of $f(x)$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447049.png" /> be an arbitrary algebraic extension. The elements of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447050.png" /> that are separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447051.png" /> form a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447052.png" />, which is the maximal separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447053.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447054.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447055.png" /> is called the separable closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447057.png" />. The degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447058.png" /> is called the separable degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447059.png" />, and the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447060.png" /> the inseparable degree, or the degree of inseparability. The inseparable degree is equal to some power of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447061.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447063.png" /> is said to be separably closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447064.png" />. In this case the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447065.png" /> is called purely inseparable. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447066.png" /> is purely inseparable if and only if
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Let $L/k$ be an arbitrary algebraic extension. The elements of the field
 
+
$L$ that are separable over $k$ form a field, $K$, which is the
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447067.png" /></td> </tr></table>
+
maximal separable extension of $k$ contained in $L$. The field $K$ is
 
+
called the separable closure of $k$ in $L$. The degree $[K:k]$ is called
that is, if any element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447068.png" /> is purely inseparable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447069.png" />. The purely inseparable extensions of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447070.png" /> form a distinguished class of extensions. If an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447071.png" /> is both separable and purely inseparable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084470/s08447072.png" />. For references see [[Extension of a field|Extension of a field]].
+
the separable degree of $L/k$, and the degree $[L:K]$ the inseparable
 +
degree, or the degree of inseparability. The inseparable degree is
 +
equal to some power of the number $p=\mathrm{char\;} k$. If $K=k$, then $k$ is said to be
 +
separably closed in $L$. In this case the extension $L/k$ is called
 +
purely inseparable. An extension $K/k$ is purely inseparable if and only
 +
if  
 +
$$K\subset k^{p^\infty} = \bigcup_n\; k^{p^{-n}},$$
 +
that is, if any element of $K$ is purely inseparable over
 +
$k$. The purely inseparable extensions of a field $k$ form a
 +
distinguished class of extensions. If an extension $K/k$ is both
 +
separable and purely inseparable, then $K=k$. For references see
 +
[[Extension of a field|Extension of a field]].

Revision as of 22:36, 17 February 2012

2020 Mathematics Subject Classification: Primary: 12Fxx [MSN][ZBL]

A separable extension of a field $k$ is an extension $K/k$ such that for some natural number $n$ the fields $K$ and $k^{p^{-n}}$ are linearly disjoint over $k$ (see Linearly-disjoint extensions). An extension that is not separable is called inseparable. Here $p$ is the characteristic of $k$. In characteristic 0 all extensions are separable.

In what follows only algebraic extensions will be considered (for transcendental separable extensions see Transcendental extension). A finite extension is separable if and only if the trace mapping ${\mathrm Tr} : K\to k$ is a non-zero function. An algebraic extension is separable if any finite subextension is separable.

The separable extensions form a distinguished class of extensions, that is, in a tower of fields $L\supset K\supset k$ the extension $L/k$ is separable if and only if $L/K$ and $K/k$ are separable; if $K_1/k$ and $K_2/k$ are separable extensions, then so is $K_1K_2/k$; for a separable extension $K/k$ and an arbitrary extension $L/k$ the extension $KL/L$ is again separable. An extension $K/k$ is separable if and only if it admits an imbedding in a Galois extension $L/k$. In this case, the number of different $k$-isomorphisms of $K$ into $L$ is the same as the degree $[K:k]$ for a finite extension $K/k$. Any finite separable extension is simple.

A polynomial $f\in k[x]$ is called separable over $k$ if none of its irreducible factors has a multiple root in an algebraic closure of $k$. An algebraic element $\def\a{\alpha}$ is called separable (over $k$) if it is a root of a polynomial that is separable over $k$. Otherwise $\a$ is called inseparable. An element $\a$ is called purely inseparable over $k$ if $\a^{p^n}\in k$ for some $n$. An irreducible polynomial $f(x)$ is inseparable if and only if its derivative $f'(x)$ is identically zero (this is possible only for $k$ of characteristic $p>0$ and $f(x)=f_1(x^p)$). An arbitrary irreducible polynomial $f(x)$ can be uniquely represented in the form $f(x)=g(x^{p^e})$, where $g(x)$ is a separable polynomial. The degree of $g(x)$ and the number $e$ are called, respectively, the reduced degree and the index of $f(x)$.

Let $L/k$ be an arbitrary algebraic extension. The elements of the field $L$ that are separable over $k$ form a field, $K$, which is the maximal separable extension of $k$ contained in $L$. The field $K$ is called the separable closure of $k$ in $L$. The degree $[K:k]$ is called the separable degree of $L/k$, and the degree $[L:K]$ the inseparable degree, or the degree of inseparability. The inseparable degree is equal to some power of the number $p=\mathrm{char\;} k$. If $K=k$, then $k$ is said to be separably closed in $L$. In this case the extension $L/k$ is called purely inseparable. An extension $K/k$ is purely inseparable if and only if $$K\subset k^{p^\infty} = \bigcup_n\; k^{p^{-n}},$$ that is, if any element of $K$ is purely inseparable over $k$. The purely inseparable extensions of a field $k$ form a distinguished class of extensions. If an extension $K/k$ is both separable and purely inseparable, then $K=k$. For references see Extension of a field.

How to Cite This Entry:
Separable extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_extension&oldid=21154
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article