The polynomial
$$
h(t)=t^3-t-1\in\mathbb F_3[t]
$$
has no root in $\mathbb F_3$ and is therefore irreducible. Define
$$
\left|\frac{a(t)}{b(t)}\right|_h
=c^{\operatorname{ord}_h(b)-\operatorname{ord}_h(a)}
$$
for any fixed $c>1$. Its valuation ring has residue field
$$
\mathbb F_3[t]/(h)\cong\mathbb F_{27}.
$$
The <completion at an irreducible polynomial over a finite field> identifies the completion with $\mathbb F_{27}((T))$, where $T$ corresponds to the uniformizer $h(t)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 2 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-136/2/b/ii/solution