Answered: #5) Same as in #4 but now regarding…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.2: Divisibility And Greatest Common Divisor

Problem 35E

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please just answer #5. Thank you

#3) Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that (-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p == +- 1 (mod 8) for odd primes p.

#4) As in #3 but now evaluate (37603|48611), again by regarding it as a Legendre symbol, after first checking by trial division (by hand - be sure to mention how far you need to go to be sure!), that 48611 is a prime and 37603 is a semiprime (product of two distinct primes).

#5) Same as in #4 but now regarding (37603|48611) as a Jacobi symbol and using the Law of Quadratic Reciprocity's generalization to Jacobi symbols - this time avoiding knowledge of the factorization of 37603 and the known primality of 48611.

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