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In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in . An integral element is a root of a monic polynomial with rational integer coefficients, . This ring is often denoted by or . Since any rational integer number belongs to and is its integral element, the ring is always a subring of . The ring is the simplest possible ring of integers.〔''The ring of integers'', without specifying the field, refers to the ring of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.〕 Namely, where is the field of rational numbers.〔Cassels (1986) p.192〕 And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this. The ring of integers of an algebraic number field is the unique maximal order in the field. == Properties == The ring of integers is a finitely-generated -module. Indeed it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as : with .〔Cassels (1986) p.193〕 The rank of as a free -module is equal to the degree of over . The rings of integers in number fields are Dedekind domains.〔Samuel (1972) p.49〕 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「Ring of integers」の詳細全文を読む スポンサード リンク
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