= Solution
The <Laurent series field> $k((t))$ consists of series $\sum_{n\geq N}a_nt^n$. The map
$$
v_t\left(\sum a_nt^n\right)=\min\{n:a_n\ne0\}
$$
is 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 $K^*=t^{\mathbb Z}\times\mathbb F_p^*\times(1+t\mathbb F_p[[t]])$. 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 $\sum a_nt^n$ to $\sum a_n^2t^{2n}$, 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.
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