Section 2.3: The Arithmetic of $latex\mathbb{Z}_p$ (p Prime) and $latex\mathbb{Z}_n$$

In my analysis of this section I noted that Hungerford often switches between the conventional modular arithmetic notation (with the brackets []) and then uses the number by itself. While both are completely valid certain uses are better than others. That is the notation [a] is much better than a if there are integers in being used at the same time. That is if b\in\mathbb{Z} and a\in\mathbb{Z}_n then ab and b\cdot [a] mean the same thing but the second notation indicates that b is not a congruence class.

Now to continue with Section 2.3:

Question:

Does this work for \mathbb{Z}_13? [a]\odot [b]=[ab]=[0] means 13\mid ab implies 13\mid a or 13\mid b.

We’ll later cover this but note that if we try to solve ax=1 in \mathbb{Z} then the only solutions for a are a=\pm 1. But the same equation in \mathbb{Z}_5 has multiple solutions: [2]\odot [3]=[1] and [4]\odot [4]=[1].

Theorem 2.8:

For a positive integer p, the following are equivalent:

  1. p is prime
  2. For non zero a\in\mathbb{Z}_p, the equation ax=1 has a solution
  3. Whenever bc=0 in $latex \mathbb{Z}_p$ then $$b=0$$ or c=0

This theorem explains a lot but fails to explain when there are solutions to the equation ax=1.

Theorem 2.9

Let $latex a,n\in\mathbb{Z}$, n>1. The equation [a]\odot x=[1] has a solution $latex \mathbb{Z}_n$ if and only if (a,n)=1.

To avoid repeating the equation [a]\odot x=1 let’s define something to indicate when we reference this:

Definition:

A non zero [a] in \mathbb{Z}_n is called a unit if [a]\odot x=[1] has a solution x\in\mathbb{Z}_n.

A solution x=[0] of [a]\odot x=[1] is called a multiplicative inverse of a.

Using this new definition let’s re-write Theorem 2.9: [a] is a unit if and only if (a,n)=1.

Definition:

A non zero element of \mathbb{Z}_n is called a zero-divisor if [a]\odot x=[0] has a non  zero solution x=[b] in \mathbb{Z}_n.

Something not mentioned in the third edition of Hungerford’s Abstract Algebra but useful nevertheless are the following facts:

For any non-zero a\in\mathbb{Z}_p (p is prime) ax=b has a unique solution.

If (a,n)=1, then ax=1 has a unique solution.

If (a,n)=d>1, then ax=b has solutions if and only if d\mid b, and in this case there are exactly d solutions.

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