Polynomial ring F_2[t]

The polynomial ring $\mathbb F_2[t]$ is the set of finite sums $\sum_{j=0}^m a_jt^j$ over the finite field $\mathbb F_2$ , with coefficientwise addition and multiplication determined by $t^it^j=t^{i+j}$ .

Ring and maps

Its units are the nonzero constants. A unital $\mathbb F_2$ -algebra map from this ring to another commutative unital $\mathbb F_2$ -algebra is determined by the chosen image of $t$ . Its fraction field is $\mathbb F_2(t)$ .

Completion is extra structure

The formal power-series ring $\mathbb F_2[\![t]\!]$ arises by $t$ -adic completion, not by merely allowing a larger finite polynomial degree. This record is algebraic and does not impose a topology.