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In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, algebraic number fields, ''p''-adic fields, and so forth. Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory 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 and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry. As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2''x'' = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a ''division ring'' or ''skew field''. (Historically, division rings 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: == Definition and illustration == Intuitively, a field is a set ''F'' that is a commutative group with respect to two compatible operations, addition and multiplication (the latter excluding zero), with "compatible" being formalized by ''distributivity'', and the caveat that the additive and the multiplicative identities are distinct (0 ≠ 1). The most common way to formalize this is by defining a ''field'' as a set together with two operations, usually called ''addition'' and ''multiplication'', and denoted by + and ·, respectively, such that the following axioms hold (note that ''subtraction'' and ''division'' are defined in terms of the inverse operations of addition and multiplication):〔That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative (its additive inverse); this is not taken as given, but is implicitly defined in terms of addition as " is the unique ''b'' such that ", "implicitly" because it is defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse. In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of "reciprocal" (multiplicative inverse).〕 ;''Closure'' of ''F'' under addition and multiplication :For 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 multiplication :For all ''a'', ''b'', and ''c'' in ''F'', the following equalities hold: ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' and ''a'' · (''b'' · ''c'') = (''a'' · ''b'') · ''c''. ;''Commutativity'' of addition and multiplication :For all ''a'' and ''b'' in ''F'', the following equalities hold: ''a'' + ''b'' = ''b'' + ''a'' and ''a'' · ''b'' = ''b'' · ''a''. ; Existence of additive and multiplicative ''identity elements'' :There 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 for all ''a'' in ''F'', ''a'' · 1 = ''a''. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. ;Existence of ''additive inverses'' and ''multiplicative inverses'' :For every ''a'' in ''F'', there exists an element −''a'' in ''F'', such that . Similarly, for any ''a'' in ''F'' other than 0, there exists an element ''a''−1 in ''F'', such that . (The elements and are also denoted and ''a''/''b'', respectively.) In other words, ''subtraction'' and ''division'' operations exist. ;''Distributivity'' of multiplication over addition :For all ''a'', ''b'' and ''c'' in ''F'', the following equality holds: . A field is therefore an algebraic structure ; of type , consisting of two abelian groups: * ''F'' under +, −, and 0; * ''F'' ∖ under ·, −1, and 1, with 0 ≠ 1, with · distributing over +.〔Wallace, D A R (1998) ''Groups, Rings, and Fields'', SUMS. Springer-Verlag: 151, Th. 2.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Field (mathematics)」の詳細全文を読む スポンサード リンク
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