I will briefly introduce the idea and development behind SAGE. Then I will do several examples that include the Chinese Remainder Theorem (Sun Tzu Theorem), and I will check quadratic reciprocity for a couple values of p. Then I plan to move to algebraic number theory and show how to compute discriminants, units, and signatures of number fields, as well as the behavior of rational primes in number fields. These examples will mostly come from some of the assignments in the current algebraic number theory course. Finally, if we have time, I will tell you how to compute the number of solutions of an equation over the simplest finite fields in two ways: the "easy" way, which is horrific for computations, and the "hard" way which uses Gauss sums.
The talk should be accessible to everyone and I will try to give most of the definitions and sketch some the ideas. However on the programming side, I will not explain the flow of control (if-statements, for-loops and so on) explicitly, but rather illustrate their use with examples.