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Topic: Quadratic reciprocity

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	Quadratic reciprocity - Wikipedia, the free encyclopedia
		The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol.
		Therefore it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extensions can rightly be considered a generalization of quadratic reciprocity to all global fields.
		There are cubic, quartic (biquadratic) and other higher reciprocity laws; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).
	en.wikipedia.org /wiki/Quadratic_reciprocity (1069 words)

	Quadratic reciprocity (Site not responding. Last check: 2007-11-03)
		In mathematics, the law of quadratic reciprocity in number theory, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic.
		There are cubic, quartic (biquadratic) and other higher reciprocity laws ; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).
		A Mechanical Proof of Quadratic Reciprocity A paper by David M. Russinoff describing the use of the Boyer-Moore theorem prover in mechanically generating a proof of the Law of Quadratic Reciprocity.
	www.serebella.com /encyclopedia/article-Quadratic_reciprocity.html (853 words)

	PlanetMath: quadratic reciprocity for polynomials
		The quadratic reciprocity theorem for polynomials over a finite field states that
		"quadratic reciprocity for polynomials" is owned by djao.
		This is version 4 of quadratic reciprocity for polynomials, born on 2002-01-17, modified 2003-09-15.
	planetmath.org /encyclopedia/QuadraticReciprocityForPolynomials.html (136 words)

	Orðasafn: Q
		quadratic equation annars stigs jafna, ferningsjafna (?), = quadratic, = quadric 2.
		quadratic reciprocity law ferningsgagnkvæmnissetning, setning um ferningsgagnkvæmni, = law of quadratic reciprocity.
		2 annars stigs jafna, ferningsjafna, = quadratic, = quadratic equation.
	www.hi.is /~mmh/ord/safn/safnQ.html (249 words)

	Quadratic reciprocity (Site not responding. Last check: 2007-11-03)
		In mathematics the law of quadratic reciprocity in number theory conjectured by Euler and Legendre and first satisfactorily proved by Gauss connects the solvability of two related equations in modular arithmetic.
		There are cubic quartic (biquadratic) and other reciprocity laws; but since two of the roots of 1 (root of unity) are not real cubic reciprocity is the arithmetic of the rational numbers (and same applies to higher laws).
		The Lemma of Gauss reasons about the of quadratic residues and is involved in Gauss's proof quadratic reciprocity.
	www.freeglossary.com /Quadratic_reciprocity (315 words)

	Quadratic residue - Wikipedia, the free encyclopedia
		In mathematics, a number q is called a quadratic residue modulo n if there exists an integer x such that:
		They occur in a rather random pattern; this has been exploited in applications to acoustics.
		In effect, a quadratic residue modulo n is a number that has a square root in modular arithmetic when the modulus is n.
	en.wikipedia.org /wiki/Quadratic_residue (239 words)

	Quadratic reciprocity (Site not responding. Last check: 2007-11-03)
		In mathematics, the law of quadratic reciprocity in number theory, conjectured by Euler and Legendre and firstsatisfactorily proved by Gauss, connects the solvability oftwo related quadratic equations in modular arithmetic.
		There are cubic, quartic (biquadratic) and other higher reciprocity laws ; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of therational numbers (and the same applies to higher laws).
		The Lemma of Gauss reasons about the properties of quadratic residues and is involved in Gauss's proof of quadraticreciprocity.
	www.therfcc.org /quadratic-reciprocity-34808.html (287 words)

	[No title]
		This extremely interesting monograph on reciprocity laws for power residues covers beautifully their development starting from the time of Euler who, in 1744, stated a result equivalent to the well-known Law of Quadratic Reciprocity (abbreviated as LQR) for distinct odd prime numbers p,q.
		Besides Herglotz's proof (via elliptic functions) for quadratic reciprocity laws in imaginary quadratic fields,one also finds herein, an account of Takagi's results on abelian extensions of the field K :=Q(i) (being realizable) as subfields of extensions of K generated by division values of elliptic functions (viewed in the light of Kronecker's Jugendtraum).
		Quadratic reciprocity laws for all algebraic number fields were established by Hilbert's generalization of LQR, with his norm residue symbol replacing the Legendre symbol in LQR.
	www.rzuser.uni-heidelberg.de /~hb3/revR.txt (884 words)

	Quadratic residue: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-03)
		In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself....
		Quadratic residues are used in the Legendre symbol Legendre symbol quick summary:
		The legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues....
	www.absoluteastronomy.com /encyclopedia/q/qu/quadratic_residue.htm (655 words)

	The Prime Glossary: quadratic residue
		In the study of diophantine equations (and surprisingly often in the study of primes) it is important to know whether the integer a is the square of an integer modulo p.
		If it is, we say a is a quadratic residue modulo p; otherwise, it is a quadratic non-residue modulo p.
		One of the most important results about quadratic residues is expressed in the surprisingly difficult to prove quadratic reciprocity theorem (see the entry on the Legendre symbol).
	primes.utm.edu /glossary/page.php?sort=QuadraticResidue (168 words)

	Quadratic Reciprocity (Site not responding. Last check: 2007-11-03)
		Quadratic reciprocity -- Facts, Info, and Encyclopedia article...
		Gauß, Eisenstein, and the ``third'' proof of the Quadratic Reciprocity Theorem:...
		"Quadratic Reciprocity in Isabelle" by David Emmanuel Gray...
	www.scienceoxygen.com /math/311.html (87 words)

	Quadratic reciprocity - QuickSeek Encyclopedia (Site not responding. Last check: 2007-11-03)
		Gauss called it the 'golden theorem' and was so fond of it that he went on to provide more than seven separate proofs over his lifetime.
		The quadratic reciprocity law can be formulated in terms of the Hilbert symbol $(a,b)_v$ where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the archimedean one and the p-adic absolute values for primes p).
		The Hilbert reciprocity law states that $(a,b)_v$ , for fixed a and b and varying v, is 1 for all but finitely many v and the product of $(a,b)_v$ over all v is 1.
	quadraticreciprocity.quickseek.com (1022 words)

	reciprocating saw (Site not responding. Last check: 2007-11-03)
		See live article Reciprocal In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1.
		See live article Reciprocal altruism Reciprocal altruism is a form of altruism in which one organism provides a benefit to another in the expectation of future reciprocation.
		See live article Reciprocity In international relations and treaties, the principle of reciprocity states that favours, benefits, or penalties, granted by one state to the citizens of another, should be returned in kind.
	www.bellmo.com /tools/reciprocating+saw (1119 words)

	Fermat's Last Theorem: Quadratic reciprocity
		The proof for the Rule of Quadratic Reciprocity can be found here.
		The set of quadratic residues is an important concept that derives from modular arithmetic.
		The idea of quadratic reciprocity builds on top of the Legendre Symbol and refers to the relationship between two odd primes.
	fermatslasttheorem.blogspot.com /2006/01/quadratic-reciprocity.html (566 words)

	Section 11.5: Quadratic Reciprocity
		Hint: This result, which is known as the Quadratic Reciprocity Theorem, is a tough one.
		Ironically, there may be more proofs of quadratic reciprocity than of any other theorem in the course.
		Thus by applying the Quadratic Reciprocity Theorem, we will be able to quickly simplify a complicated calculation.
	www.math.mtu.edu /mathlab/COURSES/holt/dnt/quadratic5.html (517 words)

	"Quadratic Reciprocity in Isabelle" by David Emmanuel Gray (Site not responding. Last check: 2007-11-03)
		Quadratic reciprocity is a deep and important result in number theory.
		In this spirit, we have formalized quadratic reciprocity in the semi-automated theorem prover Isabelle.
		In our proof of quadratic reciprocity, we derive Gauss’ Lemma from Euler’s Criterion, which states that for any odd prime p consider the set B = {a, 2a, 3a, …, {(p – 1) / 2}a}, where a T 0 (mod p).
	pressurecooker.phil.cmu.edu /Academic/Papers/quadRes.htm (3513 words)

	Quadratic Residue (Site not responding. Last check: 2007-11-03)
		It is always the case that 1 is a quadratic residue mod R; the main interest is in determining if there are any others.
		The deepest of the results regarding quadratic residues was first proven rigorously by Gauss, and is known as the Quadratic Reciprocity Law.
		Mathworld's article, Quadratic Residue includes a table giving the primes which have a given number, d, as a quadratic residue (left).
	mcraefamily.com /MathHelp/BasicNumberSquareQuadraticResidues.htm (1251 words)

	PlanetMath: calculating the Jacobi symbol (Site not responding. Last check: 2007-11-03)
		odd, we apply the quadratic reciprocity law and the fact that
		Cross-references: equation, congruent, factor, even, quadratic reciprocity, odd, integers, positive, Jacobi symbol, calculate
		This is version 1 of calculating the Jacobi symbol, born on 2004-12-29.
	www.planetmath.org /encyclopedia/CalculatingTheJacobiSymbol.html (116 words)

	Quadratic Reciprocity
		Whilst looking for a simple proof of the Law of Quadratic Reciprocity, Gauss (in 1808) discovered a lemma which now bears his name.
		Given an odd prime p and a residue a (mod p) which is prime to p, the lemma describes a necessary and sufficient condition for a to be a quadratic residue (mod p).
		In my view, one of the clearest expositions of the Law of Quadratic Reciprocity appears in "Number Theory for Beginners" by André Weil.
	wwwmaths.anu.edu.au /DoM/thirdyear/MATH3301/quad_reciprocity.html (927 words)

	[No title]
		Newsgroups: sci.math Subject: Re: "what is a reciprocity law?" Date: Sat, 26 Dec 1998 19:05:49 -0000 The law of quadratic reciprocity, proved by Gauss, is fundamental to number theory.
		The law of quadratic reciprocity states, for p and q distinct odd primes (p/q) = (q/p) unless p == q == 3 (mod 4) in which case (p/q) = -(q/p), or, put another way, (p/q)(q/p) = (-1)^((p-1)(q-1)/4).
		It would be interesting if ordinary quadratic >reciprocity could be proved in a similar way.
	www.math.niu.edu /~rusin/known-math/98/quadrecip (467 words)

	Gauß, Eisenstein, and the ``third'' proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel
		The Quadratic Reciprocity Theorem compares the quadratic character of two primes with respect to each other.
		`The reciprocity laws are the cornerstone of the latter theory'
		Hochgeehrtester Herr Hofrath, I too have devoted much effort to the study of the quadratic reciprocity law and have given four different proofs of it.
	www.math.nmsu.edu /~history/schauspiel/schauspiel.html (1888 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Should contain a proof of the quadratic reciprocity law by means of finite fields.
		320-324 a very short proof of the quadratic reciprocity law.
		6073 Reinhard Laubenbacher/David Pengelley: Gauss, Eisenstein, and the "third" proof of the quadratic reciprocity theorem.
	felix.unife.it /Root/d-Mathematics/d-Number-theory/b-Reciprocity-laws (115 words)

	The Jewel of Arithmetic: Quadratic Reciprocity
		These are called the quadratic residues (mod 7), and the remaining numbers 3,5,6 are called the non-quadratic residues.
		It's often extremely important when dealing with problems in number theory to know whether a certain prime p is a square (i.e., a quadratic residue) modulo some other particular prime q.
		It's just an abstract symbol equal to either +1 or -1 depending on whether p is or isn't a square modulo q.
	www.mathpages.com /home/kmath075.htm (1007 words)

	Doug's Expositions (Site not responding. Last check: 2007-11-03)
		Quadratic reciprocity is a startling result in elementary number theory.
		Although such a course starts with trivial-seeming results (such as negativenegative=positive), within a couple of months one reaches striking, completely nonobvious results, and quadratic* reciprocity is one of the milestones.
		In particular, they are useful for solving single constraint linear and quadratic equations in 2 variables.
	www.math.columbia.edu /~zare/expositions.html (1039 words)

	Wiley::The Fourier-Analytic Proof of Quadratic Reciprocity
		The relative quadratic case was first settled by Hecke in 1923, then recast by Weil in 1964 into the language of unitary group representations.
		The Fourier-Analytic Proof of Quadratic Reciprocity provides number theorists interested in analytic methods applied to reciprocity laws with a unique opportunity to explore the works of Hecke, Weil, and Kubota.
		This work brings together for the first time in a single volume the three existing formulations of the Fourier-analytic proof of quadratic reciprocity.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471358304,descCd-description.html?print=true (230 words)

	Number theory Article, Numbertheory Information (Site not responding. Last check: 2007-11-03)
		Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity.
		The properties of multiplicative functions such as the Möbius function and Euler's φfunction are investigated; so are integer sequences such as factorials and Fibonacci numbers.
		Besides summarizing previous work, Legendre stated the law of quadratic reciprocity.
	www.anoca.org /numbers/integers/number_theory.html (1177 words)

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