Pete's answer is very good. Some self-promotion:

An [expository post][1] on the relationship between factoring polynomials and factoring primes.

The [previous answer][2] regarding the polynomial $t^3+t^2−2t−1$


  [1]: http://sbseminar.wordpress.com/2007/08/24/factoring-polynomials-and-the-frobenius/
  [2]: http://mathoverflow.net/questions/10201/a-problem-of-shimura-and-its-relation-to-class-field-theory/10225#10225