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 is much better than a if there are integers in being used at the same time. That is if
and
then
and
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 ?
means
implies
or
.
We’ll later cover this but note that if we try to solve in
then the only solutions for a are
. But the same equation in
has multiple solutions:
and
.
Theorem 2.8:
For a positive integer p, the following are equivalent:
- p is prime
- For non zero
, the equation
has a solution
- Whenever
in $latex \mathbb{Z}_p$ then $$b=0$$ or
This theorem explains a lot but fails to explain when there are solutions to the equation .
Theorem 2.9
Let $latex a,n\in\mathbb{Z}$,
. The equation
has a solution $latex \mathbb{Z}_n$ if and only if
.
To avoid repeating the equation let’s define something to indicate when we reference this:
Definition:
A non zero in
is called a unit if
has a solution
.
A solution of
is called a multiplicative inverse of a.
Using this new definition let’s re-write Theorem 2.9: is a unit if and only if
.
Definition:
A non zero element of is called a zero-divisor if
has a non zero solution
in
.
Something not mentioned in the third edition of Hungerford’s Abstract Algebra but useful nevertheless are the following facts:
For any non-zero (p is prime)
has a unique solution.
If , then
has a unique solution.
If , then
has solutions if and only if
, and in this case there are exactly d solutions.