Perfect Field
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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., 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, 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 \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, subt ...
\mathbb_q; * 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 \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 ''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 prese ...
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 :\cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots where the transition maps are the Frobenius endomorphism. The inverse limit 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 In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exis ...
* Quasi-finite field


Notes


References

* * * * * *


External links

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