You have multiplied polynomials with real coefficients for years. The same rules work with coefficients in any ring, and polynomials over $\mathbb{Z}_{p}$ will build the finite fields. But the coefficient ring matters: over $\mathbb{Z}_{6}$ some familiar facts fail.
Definition. For a commutative ring $R$, $R[x]$ is the set of expressions $a_{n}x^{n} + \cdots + a_{1}x + a_{0}$ with $a_{i} \in R$, added and multiplied as usual, with the coefficients computed in $R$: $$\Big(\sum a_{i}x^{i}\Big)\Big(\sum b_{j}x^{j}\Big) = \sum_{k} \Big(\sum_{i + j = k} a_{i}b_{j}\Big)x^{k}.$$ The degree is the highest power with a nonzero coefficient, the leading coefficient. Over $\mathbb{Z}_{n}$ we write coefficients from $0$ to $n - 1$.
Why it works
Proof. If $f$ has leading term $a_{m}x^{m}$ and $g$ has $b_{n}x^{n}$, the highest term of $fg$ is $a_{m}b_{n}x^{m+n}$, and $a_{m}b_{n} \ne 0$ because $R$ has no zero divisors. So $fg \ne 0$ and its degree is $m + n$. $\square$ It follows that the units of $F[x]$, for a field $F$, are exactly the nonzero constants: if $fg = 1$, the degrees add to $0$.
Worked example: Multiplying in $\mathbb{Z}_{5}[x]$
- Over the integers: $6x^{3} + 3x^{2} + 9x + 8x^{2} + 4x + 12 = 6x^{3} + 11x^{2} + 13x + 12$.
- Reduce each coefficient mod $5$: $x^{3} + x^{2} + 3x + 2$.
Answer: $x^{3} + x^{2} + 3x + 2$.
Worked example: Degrees can drop
- $6x^{2} + 2x + 3x + 1 = 6x^{2} + 5x + 1$.
- $6 \equiv 0$, so the product is $5x + 1$, of degree $1$, not $2$.
Answer: $5x + 1$: the leading coefficients $2$ and $3$ are zero divisors.
Worked example: A strange unit
- $(1 + 2x)(1 + 2x) = 1 + 4x + 4x^{2} = 1$ in $\mathbb{Z}_{4}[x]$.
- So $1 + 2x$ is its own inverse, a unit of degree $1$. This cannot happen over a field.
Answer: $(1 + 2x)^{2} = 1$.
Counting. Over $\mathbb{Z}_{p}$ a polynomial of degree at most $d$ has $d + 1$ coefficients, each with $p$ choices, so there are $p^{d+1}$ of them (including $0$); exactly $p^{d}$ are monic of degree $d$. These finite counts are what make $\mathbb{Z}_{p}[x]$ such a good laboratory.
Common mistake
Key idea