# Perfect field

A field over which every polynomial is separable. In other words, every algebraic extension of is a separable extension. All other fields are called imperfect. Every field of characteristic 0 is perfect. A field of finite characteristic is perfect if and only if , that is, if raising to the power is an automorphism of . Finite fields and algebraically closed fields are perfect. An example of an imperfect field is the field of rational functions over the field , where is the field of elements. A perfect field coincides with the field of invariants of the group of all -automorphisms of the algebraic closure of . Every algebraic extension of a perfect field is perfect.

For any field of characteristic with algebraic closure , the field

is the smallest perfect field containing . It is called the perfect closure of the field in .

#### 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. L.V. Kuz'min (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Perfect_field&oldid=11515