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Group theory

Group Theory Questions. See attached file for full problem description.

( p is odd prime and a an integer not divisible by p ) the LEGENDRE SYMBOL IS DEFINRD BY = 1 if a is a quadratic residue of P, = -1 if a is a quadratic nonresidue of P.

a. Evaluate

b. Evaluate Hint 4567 is a prime.

c. Does 17 ( mod 29) have a solution ?

d. Does 19 ( mod 30) have a solution ?

e. Use Gauss's lemma to show that 17 (mod 19) is solvable.

f. Solve the congruence (mod 45).

g. Solve the congruence (mod 343).

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Solution Preview

Please see the attached file.

For Legendre Symbol, we have the following properties which will be used in the following proof.
(2) (mod )
(4) if
(5) ,
(7) Quadratic Reciprocity Theorem:
If both and are odd primes, then . In another word, if one of and is ...

Solution Summary

This is a series of questions involving Gauss's lemma, congruences, and groups.