What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

Question

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in turn making $\langle p(x) \rangle$ a maximal ideal. What exactly are the elements of this quotient ring? I have developed an intuition of thinking about quotient groups in terms of cosets, but that doesn't seem to apply as well here, at least on the intuitive level.

Answer 1

I expect you are familiar with modular arithmetic. One of the intuitive ways that you can think of arithmetic modulo $5$, say, is that it is just normal arithmetic except that $5$ is "the same as" $0$. For example, $$3\times4=12=2\times5+2=2\times0+2=2\ .$$

You can think of this in much the same way: ${\Bbb Z}_n[x]$ is the set of all polynomials with coefficients modulo $n$: for example, in ${\Bbb Z}_5[x]$ we have $$\eqalign{(1+2x+3x^2)(4+3x+2x^2) &=4+11x+20x^2+13x^3+6x^4\cr &=4+x+3x^3+x^4\ ,\cr}$$ because $5$ "is the same as" $0$. In ${\Bbb Z}_n[x]/\langle p(x)\rangle$ you go one step further and say that $p(x)$ also "is the same as" the zero polynomial. For example, continuing the above example, in ${\Bbb Z}_5[x]/\langle x^3+x+1\rangle$ we have $$\eqalign{(1+2x+3x^2)(4+3x+2x^2) &=4+x+3x^3+x^4\cr &=(x^3+x+1)(x+3)+(4x^2+2x+1)\cr &=4x^2+2x+1\ .\cr}$$ Hope this helps!