- Exponentiation
- [
*a*]^{n}is well defined,*n*^{[a]}is not.

- [
- Some elements of
*Z*_{m}have inverses- key terms: inverse, unit, relatively prime/coprime, totient
- key fact: [
*a*]_{m}has an inverse if*g**c**d*(*a*,*m*) = 1.

- Teaser for next lecture: [
*a*]_{m}^{[b]ϕ(m)}is well defined.

We quickly did the proof that multiplication in *Z*_{m} is well defined. It is very similar to the proof that addition of equivalence classes is well defined.

Proof sketch: Assume [*a*] = [*a*ʹ] and [*b*] = [*b*ʹ]. Unfolding these definitions gives *a* = *a*ʹ − *m**c* and *b* = *b*ʹ − *m**c*. Muliplying these gives *a**b* = *a*ʹ*b*ʹ + *m*(⋯). Rearranging gives us *a**b* − *a*ʹ*b*ʹ = *m*(⋯), so *m*∣*a**b* − *a*ʹ*b*ʹ, and thus [*a**b*] = [*a*ʹ*b*ʹ] as required.

Raising an equivalence class to an integer power **is** well defined.

In more detail: *e**x**p*: *Z*_{m} × *Z* → *Z*_{m} given by *e**x**p*: [*a*], *n*↦[*a*^{n}] is well defined. We could prove this directly, but it follows from the fact that raising *a* to the *n* is just multiplying *a* by itself *n* times. One can do induction on *n*; the inductive step just uses the fact that multiplication is well defined.

Raising an integer (or an equivalence class) to the power of an equivalence class is **not** well defined.

In more detail: *e**x**p*: *Z* × *Z*_{m}→*Z*_{m} given by *e**x**p*: *n*, [*a*]↦[*n*^{a}] is not well defined. For example, working mod 5, we would hope that 2^{[3]} = 2^{[8]}. But 2^{3} = 8 and 2^{8} = 256, and [8] = [3] ≠ [1] = [256].

**Summary:** [*a*]^{n} is okay, *n*^{[a]} is not.

We will recover exponentiation next lecture.

Unlike the integers, you can divide by some of the elements of *Z*_{m}.

If

*S*is a set with some (reasonable) notion of multiplication, and if*x*∈*S*, then an**inverse**of*x*is an element*y*∈*S*such that*x**y*= 1.If

*x*has an inverse, then*x*is called a**unit**

The units of

*Z*are 1 (its inverse is 1) and − 1 (its inverse is − 1).The units of

*R*(the real numbers) are all reals except 0.The units of

*Q*(the rational numbers) are all rationals except 0(if you're familiar with linear algebra) the units in the set of

*n*×*n*matrices are those with non-zero determinants.

**Important Fact:** [*a*] ∈ *Z*_{m} is a unit if and only if *g**c**d*(*a*, *m*) = 1. This is only true if *a* and *m* share no common factors (other than 1). In this case, *a* and *m* are said to be **coprime** or **relatively prime**.

You are proving this fact on the current homework.

Definition: **( Z_{m})^{ * }** is the set of units of

Examples:

(

*Z*_{5})^{ * }= {[1], [2], [3], [4]}. Note that [0] is not a unit. By inspection, the inverse of [1] is [1], the inverse of [4] is [4], and [2] and [3] are inverses of each other.More generally, if

*p*is prime, then all non-zero elements of*Z*_{p}are units, because they can't share a factor with*p*(since*p*is prime).(

*Z*_{6})^{ * }= {[1], [5]}. 2, 3, and 4 all share factors with 6, and are thus not units.[0] is never a unit. [1] is always a unit. [

*m*− 1] = [ − 1] is also always a unit (and is its own inverse).

Definition: The **totient** of *m*, written ** ϕ(m)** is the number of units of

- By examples above,
*ϕ*(5) = 4,*ϕ*(6) = 2, and*ϕ*(*p*) =*p*− 1 if*p*is prime.