Notation
\(\langle x \mid x^i x^j = x^{i+j} \rangle\): 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).