"이차잉여의 상호법칙"의 두 판 사이의 차이

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72번째 줄: 72번째 줄:
 
* [[프로베니우스와 체보타레프 밀도(density) 정리|Chebotarev density theorem]]
 
* [[프로베니우스와 체보타레프 밀도(density) 정리|Chebotarev density theorem]]
 
* [[자코비 세타함수|세타함수]]
 
* [[자코비 세타함수|세타함수]]
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79번째 줄: 81번째 줄:
 
* [http://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691124922 Fearless Symmetry: Exposing the Hidden Patterns of Numbers]<br>
 
* [http://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691124922 Fearless Symmetry: Exposing the Hidden Patterns of Numbers]<br>
 
** Avner Ash, Robert Gross
 
** Avner Ash, Robert Gross
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* [http://www.rzuser.uni-heidelberg.de/~hb3/rec.html Reciprocity Laws. From Euler to Eisenstein]<br>
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* [http://www.rzuser.uni-heidelberg.de/%7Ehb3/rec.html Reciprocity Laws. From Euler to Eisenstein]<br>
 
** Franz Lemmermeyer (Springer, 2000)
 
** Franz Lemmermeyer (Springer, 2000)
 
* [http://www.amazon.com/Fourier-Analytic-Proof-Quadratic-Reciprocity/dp/0471358304 The Fourier-Analytic Proof of Quadratic Reciprocity]<br>
 
* [http://www.amazon.com/Fourier-Analytic-Proof-Quadratic-Reciprocity/dp/0471358304 The Fourier-Analytic Proof of Quadratic Reciprocity]<br>
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<h5>참고할만한 자료</h5>
 
<h5>참고할만한 자료</h5>
  
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* [http://www.math.kth.se/%7Eakarl/langmemorial.pdf Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals]<br>
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* [http://www.math.kth.se/~akarl/langmemorial.pdf Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals]<br>
 
 
** Anders Karlsson
 
** Anders Karlsson
 
* [http://www.jstor.org/stable/2322482 Quadratic Reciprocity: Its Conjecture and Application]<br>
 
* [http://www.jstor.org/stable/2322482 Quadratic Reciprocity: Its Conjecture and Application]<br>
102번째 줄: 102번째 줄:
 
* [http://www.jstor.org/stable/2690368 Euler and Quadratic Reciprocity]<br>
 
* [http://www.jstor.org/stable/2690368 Euler and Quadratic Reciprocity]<br>
 
** Harold M. Edwards, <cite>Mathematics Magazine</cite>, Vol. 56, No. 5 (Nov., 1983), pp. 285-291
 
** Harold M. Edwards, <cite>Mathematics Magazine</cite>, Vol. 56, No. 5 (Nov., 1983), pp. 285-291
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* [http://www.jstor.org/stable/3219217 Theorems on Quadratic Residues]<br>
 
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** Albert Leon Whiteman, <cite>Mathematics Magazine</cite>, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 71-74
 
 
* [http://www.jstor.org/stable/2690080 Why Study Equations over Finite Fields?]<br>
 
* [http://www.jstor.org/stable/2690080 Why Study Equations over Finite Fields?]<br>
 
** Neal Koblitz, <cite>Mathematics Magazine</cite>, Vol. 55, No. 3 (May, 1982), pp. 144-149
 
** Neal Koblitz, <cite>Mathematics Magazine</cite>, Vol. 55, No. 3 (May, 1982), pp. 144-149
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* [http://www.jstor.org/stable/2317083 What is a Reciprocity Law?]<br>
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** B. F. Wyman, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 79, No. 6 (Jun. - Jul., 1972), pp. 571-586
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* [http://www.jstor.org/stable/3219217 Theorems on Quadratic Residues]<br>
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** Albert Leon Whiteman, <cite style="line-height: 2em;">Mathematics Magazine</cite>, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 71-74
 
* [http://ko.wikipedia.org/wiki/%EC%9D%B4%EC%B0%A8%EC%9E%89%EC%97%AC http://ko.wikipedia.org/wiki/이차잉여]
 
* [http://ko.wikipedia.org/wiki/%EC%9D%B4%EC%B0%A8%EC%9E%89%EC%97%AC http://ko.wikipedia.org/wiki/이차잉여]
 
* http://en.wikipedia.org/wiki/quadratic_reciprocity
 
* http://en.wikipedia.org/wiki/quadratic_reciprocity
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* http://viswiki.com/en/
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
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* 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=
 
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* 다음백과사전 http://enc.daum.net/dic100/search.do?q=
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]
 
  
 
 
 
 
133번째 줄: 132번째 줄:
 
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
 
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
 
* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
 
* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
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<h5>이미지 검색</h5>
 
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* http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
 
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* http://images.google.com/images?q=
 
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* [http://www.artchive.com/ http://www.artchive.com]
 
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<h5>동영상</h5>
 
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* http://www.youtube.com/results?search_type=&search_query=
 

2009년 9월 26일 (토) 04:36 판

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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