field of quotients

既定 英語（イギリス） (en-GB)

名詞

(algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b)+(a',b')=(ab'+a'b,bb'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b)≡(a',b') if and only if ab'=a'b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).

日本語の意味

「field of quotients」とは、与えられた整域に対して、その各元を整域の非零元による順序対として表現し、加算および乗算が定義される体のことであり、分数体とも呼ばれます。

このボタンはなに？

To construct the field of quotients from an integral domain, one represents each element as an equivalence class of ordered pairs (a,b) with b nonzero and defines addition by (a,b)+(a',b')=(ab'+a'b,bb') and multiplication coordinate-wise.

整域から分数体を構成するには、各元を第二成分が0でない有序対 (a,b) の同値類として表し、加法を (a,b)+(a',b')=(ab'+a'b,bb')、乗法を成分ごとに定義する。

詳細

意味（1）

(algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b)+(a',b')=(ab'+a'b,bb'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b)≡(a',b') if and only if ab'=a'b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).

fields of quotients （ plural ）