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2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol () in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.

2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol () in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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plz solve both the question with handwritten explaination i will give you upvotes.
2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol (9) in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.
Transcribed Image Text:2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol (9) in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.
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