Polynomial Ring
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$ \langle x \mid x^i x^j = x^{i+j} \rangle $: infinite cyclic group, $i,j \in \mathbb{Z} $.
$ R(+, \times) $: ring, usually a field.
$ R[ x ] = $${ r_0 + r_1 x + \cdots + r_n x^n } $: polynomial ring.
$p(x) = r_0 + r_1 x + \cdots + r_n x^n = $$\sum_i r_i x^i$: element named $p$.
Operations
$p(x) + q(x) = $$ \sum_i p_i x^i + \sum_i q_i x^i = $$ \sum_i (p_i + q_i) x^i$.
$p(x) \times q(x) = $$\sum_i p_i x^i \times \sum_i q_j x^j = $$\sum_{i,j} (p_i \times q_j) (x^i \times x^j)$.
Degree
Max power $x^n$ with non-zero $r_n$.
Degree of $p(x) + q(x)$ is max degree of $p$ and $q$.
Degree of $p(x) \times q(x)$ is sum of degrees (in integral domain).