이차잉여의 상호법칙

간단한 소개

• 이차인 합동식 \(x^2\equiv a \pmod p\) 의 해의 개수에 관련된 문제.
• 르장드르 부호

\(\left(\frac{a}{p}\right) = \begin{cases} \;\;\,0\mbox{ if } a \equiv 0 \pmod{p} \\+1\mbox{ if }a \not\equiv 0\pmod{p} \mbox{ and for some integer }x, \;a\equiv x^2\pmod{p} \\-1\mbox{ if there is no such } x. \end{cases}\)

 

 

(정리) 이차잉여의 상호법칙

홀수인 서로 다른 소수 p, q에 대하여,

 \(\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{(p-1)(q-1)/4}\)

• \(\left(\frac{p}{q}\right) = \begin{cases} +\left(\frac{q}{p}\right)\mbox{ if }p\equiv 1 \pmod{4} \mbox{ or } q \equiv 1 \pmod{4} \\-\left(\frac{q}{p}\right)\mbox{ if } p\equiv q \equiv 3 \pmod{4} \end{cases}\) 형태로 쓸 수도 있음.
  

 

 

 

 

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관련된 다른 주제들

• 가우스합
• 더 일반적인 상호법칙들(reciprocity laws) 
• 등차수열의 소수분포에 관한 디리클레 정리
• 이차 수체에 대한 디리클레 class number 공식
• Chebotarev density theorem
• 세타함수

 

관련도서 및 추천도서

• Fearless Symmetry: Exposing the Hidden Patterns of Numbers
  
  • Avner Ash, Robert Gross
• Reciprocity Laws. From Euler to Eisenstein
  
  • Franz Lemmermeyer (Springer, 2000)
• The Fourier-Analytic Proof of Quadratic Reciprocity
  
  • Michael C. Berg
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참고할만한 자료

• What is a Reciprocity Law?
  
  • B. F. Wyman
  • The American Mathematical Monthly, Vol. 79, No. 6 (Jun. - Jul., 1972), pp. 571-586
• Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals
  
  • Anders Karlsson
• Quadratic Reciprocity: Its Conjecture and Application
  
  • David A. Cox
  • The American Mathematical Monthly, Vol. 95, No. 5 (May, 1988), pp. 442-448
• Euler and Quadratic Reciprocity
  
  • Harold M. Edwards
    
  • Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 285-291
• Theorems on Quadratic Residues
  
  • Albert Leon Whiteman
  • Mathematics Magazine, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 71-74
• Why Study Equations over Finite Fields?
  
  • Neal Koblitz
  • Mathematics Magazine, Vol. 55, No. 3 (May, 1982), pp. 144-149
• http://ko.wikipedia.org/wiki/이차잉여
• http://en.wikipedia.org/wiki/quadratic_reciprocity
• http://viswiki.com/en/
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• 다음백과사전 http://enc.daum.net/dic100/search.do?q=
• 대한수학회 수학 학술 용어집

 

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