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    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 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.