Field (mathematics)

1 Field (mathematics)1 Field (mathematics)In abstract algebra, a Field is an algebraic structure with notions of addition, subtraction, multiplication, and division,satisfying certain axioms. The most commonly used fields are the Field of real numbers, the Field of complexnumbers, and the Field of rational numbers, but there are also finite fields, fields of functions, various algebraicnumber fields, p-adic fields, and so Field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Thetheory of Field extensions (including Galois theory) involves the roots of polynomials with coefficients in a Field ;among other results, this theory leads to impossibility proofs for the classical problems of angle trisection andsquaring the circle with a compass and straightedge, as well as a proof of the Abel Ruffini theorem on the algebraicinsolubility of quintic equations.

2 In modern mathematics, the theory of fields (or Field theory) plays an essential rolein number theory and algebraic an algebraic structure, every Field is a ring, but not every ring is a Field . The most important difference is thatfields allow for division (though not division by zero), while a ring need not possess multiplicative inverses. Also,the multiplication operation in a Field is required to be commutative. A ring in which division is possible butcommutativity is not assumed (such as the quaternions) is called a division ring or skew Field . (Historically, divisionrings were sometimes referred to as fields, while fields were called commutative fields .)As a ring, a Field may be classified as a specific type of integral domain, and can be characterized by the following(not exhaustive) chain of class inclusions:Commutative rings integral domains integrally closed domains unique factorization domains principal ideal domains Euclidean domains fields finite and illustrationIntuitively, a Field is a set F that is a commutative group with respect to two compatible operations, addition andmultiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) hasno multiplicative inverse (one cannot divide by 0).

3 The most common way to formalize this is by defining a Field as a set together with two operations, usually calledaddition and multiplication, and denoted by + and , respectively, such that the following axioms hold; subtractionand division are defined implicitly in terms of the inverse operations of addition and multiplication:[1]Closure of F under addition and multiplicationFor all a, b in F, both a + b and a b are in F (or more formally, + and are binary operations on F).Associativity of addition and multiplicationFor all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a (b c) = (a b) of addition and multiplicationFor all a and b in F, the following equalities hold: a + b = b + a and a b = b and multiplicative identityThere exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a +0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that forall a in F, a 1 = a.

4 For technical reasons, the additive identity and the multiplicative identity are required tobe and multiplicative inversesFor every a in F, there exists an element a in F, such that a + ( a) = 0. Similarly, for any a in F other than 0, there exists an element a 1 in F, such that a a 1 = 1. (The elements a + ( b) and a b 1 are also denoted a bField (mathematics)2and a/b, respectively.) In other words, subtraction and division operations of multiplication over additionFor all a, b and c in F, the following equality holds: a (b + c) = (a b) + (a c).Note that all but the last axiom are exactly the axioms for a commutative group, while the last axiom is acompatibility condition between the two example: rational numbersA simple example of a Field is the Field of rational numbers, consisting of the fractions a/b, where a and b areintegers, and b 0. The additive inverse of such a fraction is simply a/b, and the multiplicative inverse providedthat a 0, as well is b/a.

5 To see the latter, note thatThe abstractly required Field axioms reduce to standard properties of rational numbers, such as the law ofdistributivityor the law of commutativity and law of example: a Field with four elements + O I A B O OIAB I IOBA A ABOI B BAIO O I A B O OOOO I OIAB A OABI B OBIAIn addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. Thefollowing example is a Field consisting of four elements called O, I, A and B. The notation is chosen such that Oplays the role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted1 above). One can check that all Field axioms are satisfied. For example:A (B + A) = A I = A, which equals A B + A A = I + B = A, as required by the above Field is called a finite Field with four elements, and can be denoted F4. Field theory is concerned withunderstanding the reasons for the existence of this Field , defined in a fairly ad-hoc manner, and describing its innerstructure.

6 For example, from a glance at the multiplication table, it can be seen that any non-zero element, , I, A,and B, is a power of A: A = A1, B = A2 = A A, and finally I = A3 = A A A. This is not a coincidence, but ratherone of the starting points of a deeper understanding of (finite) (mathematics)3 Alternative axiomatizationsAs with other algebraic structures, there exist alternative axiomatizations. Because of the relations between theoperations, one can alternatively axiomatize a Field by explicitly assuming that are four binary operations (add,subtract, multiply, divide) with axioms relating these, or in terms of two binary operations (add, multiply) and twounary operations (additive inverse, multiplicative inverse), or other usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the otheroperations to be defined in terms of these basic ones, but in other contexts, such as topology and category theory, itis important to include all operations as explicitly given, rather than implicitly defined (compare topological group).

7 This is because without further assumptions, the implicitly defined inverses may not be continuous (in topology), ormay not be able to be defined (in category theory): defining an inverse requires that one be working with a set, not amore general algebraic structuresRing and Field axiomsAbelian groupRingCommutativeringSkew fieldorDivision ring Field Abelian (additive) groupstructureYes Yes Yes Yes Yes Multiplicative structureand distributivity Yes Yes Yes Yes Commutativity of multiplication No Yes No Yes Multiplicative inverses No No Yes Yes The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence ofthe binary operation " ", together with its commutativity, associativity, (multiplicative) identity element and inversesare precisely the axioms for an abelian group. In other words, for any Field , the subset of nonzero elements F \ {0},also often denoted F , is an abelian group (F , ) usually called multiplicative group of the Field .

8 Likewise (F, +) isan abelian group. The structure of a Field is hence the same as specifying such two group structures (on the same set),obeying the other algebraic structures such as rings arise when requiring only part of the above axioms. For example,if the requirement of commutativity of the multiplication operation is dropped, one gets structures usually calleddivision rings or skew elementary group theory, applied to the abelian groups (F , ), and (F, +), the additive inverse a and themultiplicative inverse a 1 are uniquely determined by direct consequences from the Field axioms include (a b) = ( a) b = a ( b), in particular a = ( 1) aas well asa 0 = can be shown by replacing b or c with 0 in the distributive propertyField (mathematics)4 HistoryThe concept of Field was used implicitly by Niels Henrik Abel and variste Galois in their work on the solvability ofpolynomial equations with rational coefficients of degree 5 or 1857 Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axiomsof a Field .

9 This construction has been frequently recalled as a contribution to the foundations of 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the fourarithmetic operations, the German word K rper, which means "body" or "corpus" (to suggest an organically closedentity), hence the common use of the letter K to denote a Field . He also defined rings (then called order ororder-modul), but the term "a ring" (Zahlring) was invented by Hilbert.[2] In 1893, Eliakim Hastings Moore calledthe concept " Field " in English.[3]In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a Field of polynomialsin modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract Field .[4] In 1910 ErnstSteinitz published the very influential paper Algebraische Theorie der K rper (English: Algebraic Theory ofFields).

10 [5] In this paper he axiomatically studies the properties of fields and defines many important Field theoreticconcepts like prime Field , perfect Field and the transcendence degree of a Field Artin developed the relationship between groups and fields in great detail during and algebraic numbersThe Field of rational numbers Q has been introduced above. A related class of fields very important in number theoryare algebraic number fields. We will first give an example, namely the Field Q( ) consisting of numbers of the forma + b with a, b Q, where is a primitive third root of unity, , a complex number satisfying 3 = 1, 1. This fieldextension can be used to prove a special case of Fermat's last theorem, which asserts the non-existence of rationalnonzero solutions to the equationx3 + y3 = the language of Field extensions detailed below, Q( ) is a Field extension of degree 2. Algebraic number fields areby definition finite Field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector , complex numbers, and p-adic numbersTake the real numbers R, under the usual operations of addition and multiplication.