Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 2 / c

The Laurent series field $k((t))$ consists of series ${\sum }_{n\ge N}{a}_n{t}^n$. The mapis a discrete valuation. A Cauchy sequence has each coefficient eventually constant, and these stabilized coefficients define its limit, proving completeness.

For $K={\mathbb{F}}_p((t))$, write $K^{\ast }=t^{\mathbb{Z}}\times {\mathbb{F}}_p^{\ast }\times (1+t{\mathbb{F}}_p[[t]])$. If $p$ is odd, Hensel's lemma makes squaring an automorphism of the last factor, so the square-class group has order four. If $p=2$, Frobenius sends $\sum a_n{t}^n$ to $\sum a_n^2{t}^{2n}$, and the classes of units with arbitrarily placed odd-degree terms give infinitely many square classes. Hence the group is finite exactly for odd $p$.

Solved by gpt-5.6-sol high.