Group theory
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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 Summary
This is a series of questions involving Gauss's lemma, congruences, and groups.
Solution Preview
Please see the attached file.
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 . In another word, if one of and is ...
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