Laurent-series field F_q((t))

The formal Laurent-series field $\mathbb F_q((t))$ consists of series $\sum_{j\ge N}a_jt^j$ , with $N\in\mathbb Z$ and $a_j\in\mathbb F_q$ , over the finite field $\mathbb F_q$ . Addition is coefficientwise and multiplication is convolution. Its topology is defined by $v_t(\sum a_jt^j)=\min\{j:a_j\ne0\}$ , with $v_t(0)=+\infty$ .

Local structure

The ring of integers is $\mathbb F_q[\![t]\!]$ , the maximal ideal is generated by $t$ , and reduction takes the constant coefficient. With $|x|=q^{-v_t(x)}$ , this is a complete locally compact nondiscrete field of characteristic $p$ .

Formal rather than analytic

Only finitely many negative exponents are allowed. These are formal series, with convergence governed by powers of $t$ , not functions evaluated at a complex number. Every nonzero series is a power of $t$ times a unit, which supplies its inverse.

References

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