Can the concept of congruence be applied to the remainder of a polynomial division?

Question

I know this is a very simple question, so please I apologize but I am not familiar with it:

Can the concept of (modular arithmetic) congruence be applied to the remainder of a polynomial division? (e.g. when the divisor is a polynomial of at least degree 1?

For instance, $\frac{x^2-1}{x+1}=x-1$ with remainder = $0$. Then:

Does it means that $x^2-1 \equiv 0 \pmod {x+1}?$

If so, what happens if the remainder is not an integer? If I am not wrong the remainder of the polynomial division can be for instance a fraction as well so I am not sure what happens (what is the meaning) when the remainder is not exactly an integer.

I have seen that there is a tag in MSE about polynomial congruences and did some searching but the questions are very few and different, and some of them not related with a question regarding a division in which the divisor is also a polynomial.

If somebody could provide me a basic explanation as a starting point (or an online resource to learn a minimum base) would be very appreciated.

Thank you!

(If the question can be rewritten to express it better please feel free to modify it, if it is duplicated I will remove it, just let me know)

Answer 1

You were probably taught that two integers $a$ and $b$ are congruent modulo another integer $n$ if $n$ divides $a-b$. Well, the same applies to many other objects, such as polynomials but also matrices for example, which can be added and multiplied (the mathematical term is, "in any ring").

Two polynomials $P$ and $Q$ (with, say, integer or real or complex coefficients) are congruent modulo another polynomial $N$ if $N$ divides $P-Q$. In the special case where $Q = 0$, this means that $N$ divides $P$, which is what you have here.

Moreover, polynomials with (say) rational, real or complex coefficients have an Euclidean division which satisfies the same properties as the Euclidean division over the integers, which means that in turn those polynomials exhibit all the same properties you know in the integers (such as the existence of GCDs, unique factorisation, etc.). This eliminates the problem where the remainder is a fraction, since a fraction is a rational (or real, or complex, etc.) number just as well.

Answer 2

This involves possibly the topic of polynomial ring. See here: users.math.msu.edu/users/jhall/classes/codenotes/PolyAlg.pdf. A more advanced topic about this is ideals and varieties. A nice book about it can be found here: cs.amherst.edu/~dac/iva.html.