- Lemma 12: Let be prime, and let be a natural number. Then . This was a routine calculation using our earlier observation that .
- Proposition 13: Let be a natural number. Then . We proved that is a multiplicative function, and then computed .
- Theorem 14 (Lagrange): Let be a prime, and let be a polynomial with integer coefficients, with not divisible by . Then the polynomial congruence has at most solutions. We proved this by induction, using the fact that is an integral domain.
I gave out the first examples sheet, which is available on the course page.
Understanding today’s lecture
What examples of multiplicative functions can you come up with?
Can you prove an analogue of Theorem 14 in a general integral domain?
Davenport (The Higher Arithmetic) discusses some of the above topics in the way that I’ve presented them, but not all. Baker (A concise introduction to the theory of numbers), Koblitz (A Course in Number Theory and Cryptography), and Jones and Jones (Elementary Number Theory) all also discuss these topics, although again not necessarily following exactly the approach that I’ve used.
Preparation for Lecture 4
We’d like to understand the structure of (where is prime). Can you make any conjectures on the basis of experimenting with some small values of (building on what we did at the end of the lecture)? Can you prove your conjectures?
We’re going to move on to study — you could start to think about those too.