In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 k is said to be perfect if any one of the following equivalent conditions holds:

• Every irreducible polynomial
  Irreducible polynomial
  In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
  
   over k has distinct roots.
• Every polynomial
  Polynomial
  In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
  
   over k is separable
  Separable polynomial
  In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
  
  .
• Every finite extension of k is separable
  Separable extension
  In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
  
  . (This implies that every algebraic extension
  Algebraic extension
  In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
  
   of k is separable.)
• Either k has characteristic
  Characteristic (algebra)
  In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
  
   0, or, when k has characteristic p > 0, every element of k is a pth power.
• Every element of k is a qth power. (Here, q is the characteristic exponent, equal to 1 if k has characteristic 0, and equal to p if k has characteristic p > 0).
• The separable closure of k is algebraically closed.
• Every k-algebra
  Algebra (ring theory)
  In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
  
   A is a separable algebra
  Separable algebra
  A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.- Definition and First Properties :Let K be a field...
  
  ; i.e., is reduced
  Reduced ring
  In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
  
   for every field extension
  Field extension
  In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
  
   F/k.

Otherwise, k is called imperfect.

In particular, every field of characteristic zero and finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s are perfect.

More generally, a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 of characteristic p (p a prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

) is called perfect if the Frobenius endomorphism

Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...

 is an automorphism

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

. (This is equivalent to the fourth above condition "every element of k is a pth power" for integral domains.)

Examples

Examples of perfect fields are: a field of characteristic zero, finite fields, algebraically closed fields, the union of perfect fields, fields algebraic over a perfect field. (In particular, an imperfect field is necessarily transcendental over its prime subfield, which is perfect.) On the other hand, if has a positive characteristic, then , indeterminate, is not perfect. In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry.

Perfect closure and perfection

The first condition says that, in characteristic p, a field adjoined with all p-th roots (usually denoted by ) is perfect; it is called the perfect closure, denoted by . Equivalently, the perfect closure is a maximal purely inseparable subextension. If is a normal finite extension, then .

In terms of universal properties

Universal property

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

, the perfect closure of a ring A of characteristic p is a perfect ring A_p of characteristic p together with a ring homomorphism

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

 u : A → A_p such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : A_p → B such that v factors through u (i.e. v = fu). The perfect closure always exists.

The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system
where the transition maps are the Frobenius endomorphism. The inverse limit

Inverse limit

In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

of this system is R(A) and consists of sequences (x₀, x₁, ... ) of elements of A such that for all i. The map θ : R(A) → A sends (x_i) to x₀.