Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 136 / 2

One strong form of Hensel lemma is this: if $K$ is complete for a discrete valuation $v$, $f\in {\mathcal{O}}_K[X]$, and $a_0\in {\mathcal{O}}_K$ satisfiesthen there is a unique root $\alpha$ satisfying

Define the Newton iteration over a valued fieldTaylor expansion and the ultrametric inequality show inductively that $v(f^{\prime }(a_n))=v(f^{\prime }(a_0))$ andThus the valuations of the corrections $a_{n+1}-a_n$ tend to infinity, so $(a_n)$ is a Cauchy sequence. Completeness gives a limit $\alpha$, and continuity gives $f(\alpha )=0$. If $\beta$ is another root in the stated ball, Taylor expansion of $f(\beta )-f(\alpha )$ shows that the linear term has strictly smaller valuation than all higher terms unless $\beta =\alpha$, proving uniqueness.

Every $x\in {\mathbb{Q}}_2^{\times }$ has a unique form $2^{n}u$ with $n\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_2^{\times }$. Its square class first records $n\mathrm{mod}2$. An odd 2-adic unit is a square precisely when it is congruent to $1$ modulo $8$: necessity follows by squaring an odd integer, and sufficiency follows from Hensel lemma applied in its standard $2$-adic square-root form.

The odd residues $1,3,5,7$ modulo $8$ therefore give four unit square classes. Together with valuation parity this yieldsFor example, the classes of $-1$, $2$, and $5$ form a basis of the square-class group of the 2-adic numbers.

The smallest integer isIndeed, reduction modulo $p$ cannot work: the map $x\mapsto x^p$ is the identity on ${\mathbb{F}}_p^{\times }$, although not every element of ${\mathbb{Z}}_p^{\times }$ is a $p$th power.

For odd $p$, the p-adic unit group decomposes asRaising to the $p$th power is an automorphism on ${\mu }_{p-1}$. On the principal units, the p-adic logarithm identifies it with multiplication by $p$ on $p{\mathbb{Z}}_p$, soHence whether a unit is a $p$th power is determined exactly by its residue modulo $p^2$. Equivalently,This is the pth-power criterion for p-adic units.