Number Theory: In Context and Interactive

Summary: The Group of Units and Euler's Function

1. After an example revisiting solving linear congruences, we introduce The group of units in Definition 9.1.2. Yes, we check it is a group.

2. In Definition 9.2.1 the Euler $\varphi$ function is introduced, along with the incredibly important Euler's Theorem about powers of a number modulo $n\text{.}$

3. We then use Euler's Theorem in Section 9.3 to do computations with Inverses and the Chinese Remainder Theorem.

4. Explore! In Section 9.4 you are encouraged to think about not just a formula for $\varphi \text{,}$ but more sophisticated properties thereof.

5. In the last major section of this chapter, we then prove the most important formulas.

   • In Fact 9.5.1 we get a formula for $\varphi \left(p^e\right)\text{.}$

   • In Fact 9.5.2 we see that $\varphi$ is multiplicative, which should allow for a general formula in Exercise 9.6.11.

   • In Fact 9.5.4 there is a remarkable addition formula.