Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.

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• the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.

  Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

• Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.

  Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

• Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.

  Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

• Cryptography, congruences, and primes

  According to the Chinese Remainder Theorem, the solution of the system of congruence equations is (mod ) Therefore, we get (mod ) (b) The condition in (a) that and are relatively prime is necessary.

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  Let's use the quadratic reciprocity theorem. Theorem. The Quadratic Reciprocity Theorem. If p and q are odd primes and (please see the attached file) then (please see the attached file).

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  For Legendre Symbol, we have the following properties which will be used in the following proof. (1) (2) (mod ) (3) (4) if (5) , (6) (7) Quadratic Reciprocity Theorem: If both and are odd primes, then .

• Discrete Mathematics Definitions : Algorithm, Searching algorithm, Greedy algorithm, Composite, Prime, Relatively prime integers, Matrix, Matrix addition, Symmetric, Fundamental Theorem of Arithmetic

  Fundamental Theorem of Arithmetic: 11. Euclidean Algorithm: 12. Chinese Remainder Theorem: Please see the attached file for the complete solution. Thanks for using BrainMass.

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  For x,y Z given by the extended Euclidean Algorithm, such that xp +yq =1 We have Z= (xpa + yqa ), and Z= (xpa - yqa ) Use Lagrange's Theorem Please see the attachment.

• Theory of numbers

  87083 Quadratic Congruences - Theory of Numbers Quadratic Congruences. See attached file for full problem description. Let p be an odd prime. Complete the proof of the question "For which odd primes p is LS(2, p) = 1?"