In
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, a
field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every
irreducible polynomial over ''k'' has distinct roots.
* Every
irreducible polynomial over ''k'' is
separable.
* Every
finite extension of ''k'' is
separable.
* Every
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field e ...
of ''k'' is separable.
* Either ''k'' has
characteristic 0, or, when ''k'' has characteristic , every element of ''k'' is a
''p''th power.
* Either ''k'' has
characteristic 0, or, when ''k'' has characteristic , the
Frobenius endomorphism is an
automorphism of ''k''.
* The
separable closure of ''k'' is
algebraically closed.
* Every
reduced commutative
''k''-algebra ''A'' is a
separable algebra; i.e.,
is
reduced for every
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''F''/''k''. (see below)
Otherwise, ''k'' is called imperfect.
In particular, all fields of characteristic zero and all
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s are perfect.
Perfect fields are significant because
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory t ...
over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
Another important property of perfect fields is that they admit
Witt vectors.
More generally, a
ring of characteristic ''p'' (''p'' a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
) is called perfect if the
Frobenius endomorphism is an
automorphism. (When restricted to
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s, this is equivalent to the above condition "every element of ''k'' is a ''p''th power".)
Examples
Examples of perfect fields are:
* every field of characteristic zero, so
and every finite extension, and
;
* every
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
;
* every
algebraically closed field;
* the union of a set of perfect fields totally ordered by extension;
* fields algebraic over a perfect field.
Most fields that are encountered in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic . Every imperfect field is necessarily
transcendental
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
over its
prime subfield
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
(the minimal subfield), because the latter is perfect. An example of an imperfect field is the field
, since the Frobenius sends
and therefore it is not surjective. It embeds into the perfect field
:
called its perfection. Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example, consider