Section17.7Exercises

  1. Evaluate the Legendre symbols for \(p=11\) and \(a=2,3,5\) using the calculation of Eisenstein's above.
  2. Use the previous problem, your knowledge of \(\left(\frac{-1}{11}\right)\) and of perfect squares to evaluate the other Legendre symbols for \(p=11\).
  3. Make up several hard-looking Legendre symbols \(\left(\frac{a}{29}\right)\) (modulo \(p=29\)) that are easy to solve by adding \(p\) or by factoring \(a\).
  4. Use the multiplicative property of the Legendre symbol to give a congruence condition for when \(\left(\frac{-2}{p}\right)=\pm 1\).
  5. For \(0<a,b<p\), prove that at least one of \(a,b,\) and \(ab\) is a quadratic residue of \(p\).
  6. Prove that for any prime \(p\), if \(1<i,j<\frac{p}{2}\) and \(i\neq j\), then \(i^2\not\equiv j^2\) mod (\(p\)). (Hint: factor!)
  7. In the previous exercises we proved, for an odd prime \(p\), \[\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)=0\, .\] Conjecture (and, if you can, prove) a similar result for \[\sum_{a\in Q_p} a\, .\]
  8. Prove: if \(p\equiv 3\) mod (4), and if \(a\not\equiv \pm 1,0\), then \(a\) is a QR modulo \(p\) if and only if \(p-a\) is not a QR.
  9. Verify the previous exercise for \(p=23\).
  10. Let \(p\) be a prime of the form \(p=2q+1\), where \(q\) is prime (recall that \(p\) is called a Germain prime in this case). Show that every residue from 1 to \(p-2\) is either a primitive root of \(p\) or a quadratic residue. (Hint: Use Euler's criterion, and ask yourself how many possible orders an element of \(U_p\) can have.)
  11. Prove that if \(\left(\frac{2}{n}\right)\) is the Jacobi symbol instead of the Legendre symbol, it is still true that it \(=1\) precisely when \(n\equiv \pm 1\text{ mod }(8)\). (Remember, \(n\) has to be odd by definition.)
  12. Show that if \(p\) is an odd prime, then there are exactly \(\frac{p-1}{2}-\phi(p-1)\) residues which are neither QRs nor primitive roots. (Hint: don't think too hard - just do the obvious counting up.)
  13. Evaluate five non-obvious Legendre symbols \((\frac{a}{p})\) for \(p=47\) using quadratic reciprocity.
  14. Find congruence criteria for \(p\) for when \(a\in Q_p\) for \(a=-3,6\), and \(9\). (Hint: Don't do any extra work - use what you know!)