请输入您要查询的字词:

 

单词 ChineseRemainderTheoremInTermsOfDivisorTheory
释义

Chinese remainder theorem in terms of divisor theory


In a ring with a divisor theory, a congruenceMathworldPlanetmathαβ(mod𝔞)  with respect to a divisorMathworldPlanetmathPlanetmath module (http://planetmath.org/Congruences) 𝔞 that  𝔞α-β.

Theorem.  Let 𝒪 be an integral domainMathworldPlanetmath having the divisor theory  𝒪*𝔇.  For arbitrary pairwise coprime divisors 𝔞1,,𝔞s  in 𝔇 and for arbitrary elements  α1,,αs  of the domain 𝒪 there exists an element ξ in 𝒪 such that

{ξα1(mod𝔞1)    ξαs(mod𝔞s)

Proof.  Let

𝔟i:=ji𝔞j(i=1,,s).

Apparently, the divisors  𝔟1,,𝔟s  are mutually coprime, whence there are in the ring 𝒪 the elements  β1,,βs  divisible by  the divisors  𝔟1,,𝔟s,  respectively, such that

β1++βs=1.(1)

For every  ij,  the divisor 𝔞i divides 𝔟j and therefore also the element βj.  Then the equation (1) implies that  βi1(mod𝔞i) and thus the element

ξ:=α1β1++αsβs

satisfies

ξαiβiαi(mod𝔞i)

for each  i=1,,s.  Q.E.D.

References

  • 1 М. М. Постников:Введение  в  теорию  алгебраических чисел.  Издательство ‘‘Наука’’. Москва (1982).
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 21:16:34