Quadratic Reciprocity

    

Proposition (Quadratic Reciprocity) Let and be distinct odd primes, then

Proposition (Quadratic Reciprocity -Alternate Form) Let and be distinct odd primes, then

Example (Quadratic Reciprocity) Evaluate the following Legendre symbols

(a)

    Solution. Since we have Since and we have




(b)

    Solution. To evaluate we note that So we have and since we see that



Finally since and we have




(c)

    Solution.  Since we have



So we break these down into cases as follows

  since

  since

  since

  since

  since

  since

  since

since
and

since

since

















Therefore,

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Cite this as:
Quadratic Reciprocity
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/quadratic-reciprocity.html