Here's an excerpt from abstract algebra book that I'm reading and my question is given later:
The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Given
$a(x)$ with coefficients in $F$: if $x$ is regarded merely as a placeholder, then $a(x)$ is a polynomial;
if $x$ is allowed to assume values in $F$, then $a(x)$ is a polynomial function.
Then it goes:
Remember that two polynomials $a(x)$ and $b(x)$ are equal iff corresponding coefficients are equal, whereas
two functions $a(x)$ and $b(x)$ are equal $a(x) = b(x)$ for every $x$ in their domain. These two notions of
equality do not always coincide!
For example, consider the following two polynomials in $\mathbb{Z}_5[x]$:
$$
a(x) = x^5 + 1
$$
$$
b(x) = x – 4
$$
You may check that $a(0) = b(0), a(1) = b(1), \ldots, a(4) = b(4)$, hence $a(x)$ and $b(x)$ are equal functions
from $\mathbb{Z}_5$ to $\mathbb{Z}_5$.
My question: can anyone tell me why and how $a(0)=b(0)$ for the above two functions?
Best Answer
$a(0) = (0)^5 + 1 \equiv 1$ mod $5$.
$b(0) = (0) - 4 = -4 \equiv1$ mod $5$
Remember numbers in $\mathbb{Z}_5$ are the same if they differ by a multiple of 5.