Section16.8Exercises

  1. Prove that if \(e>1\), then there is no solution to \[x^2\equiv -1\text{ mod }(2^e)\; ;\] use our knowledge of squares modulo 4.
  2. For what \(n\) does \(-1\) have a square root modulo \(n\)? (Hint: prime factorization and the previous problem.)
  3. Clearly \(4\) has a square root modulo \(7\). Find all square roots of \(4\) modulo \(7^3\) without using Sage or trying all \(343\) possibilities.
  4. Solve \(x^2+3x+5\text{ mod }(15)\) using completion of squares and trial and error for square roots to confirm the computation above.
  5. Solve the following congruences without using a computer:
    • \(x^2+6x+5\) mod (17)
    • \(5x^2+3x+1\) mod (17)
  6. Prove that if \(p\) is an odd prime \[\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)=0\, .\]
  7. Show that a quadratic residue can't be a primitive root if \(p>2\).
  8. Use Euler's Criterion to find all quadratic residues of 13.
  9. Use Euler's Criterion to prove that \(2\) has a square root modulo \(p\) if \(p\equiv 1\text{ mod }(8)\).
  10. Evaluate all Legendre symbols for \(p=7\) using Euler's Criterion.