Formal power series over F_5

The formal power-series ring $\mathbb F_5[\![t]\!]$ consists of all series $\sum_{j\ge0}a_jt^j$ over the finite field $\mathbb F_5$ , with coefficientwise addition and convolution multiplication. It carries the $t$ -adic topology, whose neighborhoods of zero are the ideals $t^m\mathbb F_5[\![t]\!]$ .

Units, residue and completion

A series is invertible exactly when its constant coefficient is nonzero. This compact complete ring has maximal ideal $(t)$ and residue field $\mathbb F_5$ . It is the inverse limit of $\mathbb F_5[t]/(t^m)$ ; inversion of $t$ gives $\mathbb F_5((t))$ .

Polynomial comparison

The polynomial ring $\mathbb F_5[t]$ is dense here, but it contains only finite sums. Infinite positive tails belong to the power-series ring and generally are not rational functions.

References

J. S. Milne, Algebraic Number Theory, Chapter 7, Theorem 7.23; Propositions 7.26, 7.46; Remark 7.49, pp. 115–127.