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A [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720401.png" /> over which every polynomial is separable. In other words, every algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720402.png" /> is a [[Separable extension|separable extension]]. All other fields are called imperfect. Every field of characteristic 0 is perfect. A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720403.png" /> of finite characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720404.png" /> is perfect if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720405.png" />, that is, if raising to the power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720406.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720407.png" />. Finite fields and algebraically closed fields are perfect. An example of an imperfect field is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720408.png" /> of rational functions over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p0720409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204010.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204011.png" /> elements. A perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204012.png" /> coincides with the field of invariants of the group of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204013.png" />-automorphisms of the algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204015.png" />. Every algebraic extension of a perfect field is perfect.
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A [[Field|field]] $k$ over which every polynomial is separable. In other words, every algebraic extension of $k$ is a [[Separable extension|separable extension]]. All other fields are called ''imperfect''. Every field of characteristic 0 is perfect. A field $k$ of finite characteristic $p$ is perfect if and only if $k = k^p$, that is, if raising to the power $p$ is an automorphism of $k$. [[Finite field]]s and [[algebraically closed field]]s are perfect. An example of an imperfect field is the field $\mathbb{F}_q(X)$ of rational functions over the field $\mathbb{F}_q$, where $\mathbb{F}_q$ is the field of $q = p^n$ elements. A perfect field $k$ coincides with the field of invariants of the group of all $k$-automorphisms of the algebraic closure $\bar k$ of $k$. Every algebraic extension of a perfect field is perfect.
  
For any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204016.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204017.png" /> with algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204018.png" />, the field
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For any field $k$ of characteristic $p>0$ with algebraic closure $\bar k$, the field
 
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$$
<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/p/p072/p072040/p07204019.png" /></td> </tr></table>
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k^{p^{-\infty}} = \bigcup_n k^{p^{-n}} \subset \bar k
 
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$$
is the smallest perfect field containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204020.png" />. It is called the perfect closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072040/p07204022.png" />.
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is the smallest perfect field containing $k$. It is called the ''perfect closure'' of the field $k$ in $\bar k$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algèbre" , Masson  (1981)  pp. Chapts. 4–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algèbre" , Masson  (1981)  pp. Chapts. 4–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR></table>

Latest revision as of 21:49, 11 October 2014

A field $k$ over which every polynomial is separable. In other words, every algebraic extension of $k$ is a separable extension. All other fields are called imperfect. Every field of characteristic 0 is perfect. A field $k$ of finite characteristic $p$ is perfect if and only if $k = k^p$, that is, if raising to the power $p$ is an automorphism of $k$. Finite fields and algebraically closed fields are perfect. An example of an imperfect field is the field $\mathbb{F}_q(X)$ of rational functions over the field $\mathbb{F}_q$, where $\mathbb{F}_q$ is the field of $q = p^n$ elements. A perfect field $k$ coincides with the field of invariants of the group of all $k$-automorphisms of the algebraic closure $\bar k$ of $k$. Every algebraic extension of a perfect field is perfect.

For any field $k$ of characteristic $p>0$ with algebraic closure $\bar k$, the field $$ k^{p^{-\infty}} = \bigcup_n k^{p^{-n}} \subset \bar k $$ is the smallest perfect field containing $k$. It is called the perfect closure of the field $k$ in $\bar k$.

References

[1] N. Bourbaki, "Elements of mathematics. Algèbre" , Masson (1981) pp. Chapts. 4–5
[2] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
How to Cite This Entry:
Perfect field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perfect_field&oldid=11515
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