Theorems in number theory | Quadratic residue | Algebraic number theory | Modular arithmetic | Number theory
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Law of quadratic reciprocity — Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its , allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo . However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case using Euler's criterion one can give an explicit formula for the "square roots" modulo of a quadratic residue , namely, indeed, This formula only works if it is known in advance that is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss, who referred to it as the "fundamental theorem" in his Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151) Privately, Gauss referred to it as the "golden theorem". He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included , together with short proofs of the law's supplements (the Legendre symbols of −1 and 2). Generalizing the reciprocity law to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program. (Wikipedia).
Number Theory | Quadratic Reciprocity
We prove the quadratic reciprocity theorem. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Number Theory
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What is the formula for a perfect square trinomial and how does the discriminant fit in
👉 Learn all about the discriminant of quadratic equations. A quadratic equation is an equation whose highest power on its variable(s) is 2. The discriminant of a quadratic equation is a formula which is used to determine the type of roots (solutions) the quadratic equation have. The disc
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👉Learn how to factor quadratics using the difference of two squares method. When a quadratic contains two terms where each of the terms can be expressed as the square of a number and the sign between the two terms is the minus sign, then the quadratic can be factored easily using the diffe
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From playlist Discriminant of a Quadratic Equation | Learn About
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This lecture is part of an online undergraduate course on the theory of numbers. We state and law of quadratic reciprocity for Legendre symbols, and prove it using Gauss sums. As applications we show how to use it to calculate Legendre symbols and to test Fermat numbers to see if they are
From playlist Theory of numbers
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From playlist Introduction to number theory (Berkeley Math 115)
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Quadratic Reciprocity Examples -- Number Theory 24
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolp
From playlist Number Theory v2
Quadratic Reciprocity Examples — Number Theory 24
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From playlist Number Theory
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