Principal unit

For a non-Archimedean local field $K$ with maximal ideal ${\mathfrak{m}}_K$, a principal unit is an element of $1+{\mathfrak{m}}_K$. The higher principal-unit groups are $1+{\mathfrak{m}}_K^n$.

In a complete discrete valuation ring of residue characteristic $p$, the power map $u\mapsto u^m$ is a bijection on $1+\mathfrak{m}$ when $p\nmid m$. For a principal unit $a$, the Hensel lemma lifts the simple residue root one of $T^m-a$ uniquely. This can absorb principal-unit factors in tame radical descriptions of local fields.

For a non-Archimedean local field $K$, the $n$th higher principal-unit group is $1+{\mathfrak{m}}_K^n$. These groups form a decreasing filtration of the principal units.

For $r\ge 1$, the map $1+{\pi }^{r}a\mapsto \bar{a}$ is a surjective group homomorphism to the additive residue field with kernel $U_{r+1}$. The extra product term is divisible by ${\pi }^{r+1}$. Thus each quotient has cardinality $|k|$, a useful count when splitting the unit group by roots of unity.

The cube of $1+3t$ is $1+9(t+3t^2+3t^3)$. Conversely the polynomial $T+3T^2+3T^3-a$ has a unique simple residue root modulo three for each $a\in {\mathbb{Z}}_3$, so Hensel lemma produces a cube root of every $1+9a$. Thus the principal units modulo their cubes have order three. The change of variable removes the derivative obstruction that prevents direct simple-root lifting for $X^3-u$.

For a finite extension $K/{\mathbb{Q}}_p$ and all sufficiently large $r$, the p-adic logarithm and p-adic exponential are inverse group isomorphisms