# Difference between revisions of "Perfect field"

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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 | + | 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 $k$ of characteristic $p>0$ with algebraic closure $\bar k$, the field | For any field $k$ of characteristic $p>0$ with algebraic closure $\bar k$, the field | ||

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k^{p^{-\infty}} = \bigcup_n k^{p^{-n}} \subset \bar k | 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 $\bar k$. | + | 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=33557