Chapter 9 The Group of Units and Euler's Function
Summary: The Group of Units and Euler's Function
This chapter uses the groups viewpoint of Chapter 8 to introduce the important topic of units.
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.
In Definition 9.2.1 the Euler \(\phi\) function is introduced, along with the incredibly important Euler's Theorem about powers of a number modulo \(n\text{.}\)
We then use Euler's Theorem in Section 9.3 to do computations with Inverses and the Chinese Remainder Theorem.
Explore! In Section 9.4 you are encouraged to think about not just a formula for \(\phi\text{,}\) but more sophisticated properties thereof.
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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 \(\phi\left(p^e\right)\text{.}\)
In Fact 9.5.2 we see that \(\phi\) 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.
There are many computational Exercises, and we especially encourage trying to explore enough to make conjectures like in Exercise 9.6.18. Finally, in Section 9.7 we finally solve the questions originally raised in Question 1.1.1.