Perfect Field
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In
algebra Algebra () is one of the 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 mathematics. Elementary a ...
, a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k'' is perfect if any one of the following equivalent conditions holds: * Every
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
over ''k'' has distinct roots. * Every
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
over ''k'' is separable. * Every
finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...
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 ext ...
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 In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
is an automorphism of ''k''. * The
separable closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of ''k'' is
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. * Every reduced commutative ''k''-algebra ''A'' is a
separable algebra In mathematics, 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 A ring homomorphism (of unital, but not necessari ...
; i.e., A \otimes_k F 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, subtr ...
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 to ...
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 vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of ord ...
s. More generally, a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
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 ways ...
) is called perfect if the
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
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 set ...
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 \mathbb and every finite extension, and \mathbb; * 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, subtr ...
\mathbb_q; * every
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
; * 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 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 \mathbf_q(x), since the Frobenius sends x \mapsto x^p and therefore it is not surjective. It embeds into the perfect field :\mathbf_q(x,x^,x^,\ldots) 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 f(x,y) = x^p + ay^p \in k ,y/math> for k an imperfect field of characteristic p and ''a'' not a ''p''-th power in ''f''. Then in its algebraic closure k^ ,y/math>, the following equality holds: : f(x,y) = (x + b y)^p , where ''b'' = ''a'' and such ''b'' exists in this algebraic closure. Geometrically, this means that f does not define an affine plane curve in k ,y/math>.


Field extension over a perfect field

Any
finitely generated 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 ...
''K'' over a perfect field ''k'' is separably generated, i.e. admits a separating transcendence base, that is, a transcendence base Γ such that ''K'' is separably algebraic over ''k''(Γ).


Perfect closure and perfection

One of the equivalent conditions says that, in characteristic ''p'', a field adjoined with all ''p''-th roots () is perfect; it is called the perfect closure of ''k'' and usually denoted by k^. The perfect closure can be used in a test for separability. More precisely, a commutative ''k''-algebra ''A'' is separable if and only if A \otimes_k k^ is reduced. In terms of universal properties, the perfect closure of a ring ''A'' of characteristic ''p'' is a perfect ring ''Ap'' of characteristic ''p'' together with a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
such that for any other perfect ring ''B'' of characteristic ''p'' with a homomorphism there is a unique homomorphism such that ''v'' factors through ''u'' (i.e. ). The perfect closure always exists; the proof involves "adjoining ''p''-th roots of elements of ''A''", similar to the case of fields. 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 such that for any perfect ring ''B'' of characteristic ''p'' equipped with a map , there is a unique map such that ''φ'' factors through ''θ'' (i.e. ). The perfection of ''A'' may be constructed as follows. Consider the
projective system In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
:\cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots where the transition maps are the Frobenius endomorphism. The
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of this system is ''R''(''A'') and consists of sequences (''x''0, ''x''1, ... ) of elements of ''A'' such that x_^p=x_i for all ''i''. The map sends (''xi'') to ''x''0., section 4.2


See also

* p-ring * Perfect ring * Quasi-finite field


Notes


References

* * * * * *


External links

* {{springer, title=Perfect field, id=p/p072040 Ring theory Field (mathematics)