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For $K={\mathbb{Q}}_p$ and $\pi =p$, choose the Lubin–Tate series $f(T)=(1+T{)}^p-1$. The associated formal group law is the formal multiplicative group. Its torsion points are $\zeta -1$ for ${\zeta }^{p^n}=1$, so its Lubin–Tate extension is the cyclotomic extension of a p-adic field ${\mathbb{Q}}_p({\mu }_{p^n})$. The Galois action $\zeta \mapsto {\zeta }^a$ gives $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, including the trivial first layer when $p=2$.

Let $L_n={\mathbb{Q}}_2({\xi }_{2^n})$. The extension is a cyclotomic extension of a p-adic field and is totally ramified. Under Local Artin reciprocity, the action on ${\xi }_{2^n}$ is the reduction of a 2-adic unit modulo $2^n$, up to the harmless inverse determined by the Frobenius convention. A uniformizer acts trivially because $L_n/{\mathbb{Q}}_2$ has no unramified part. The kernel is thereforeBy the norm subgroup of a local field extension, this kernel is exactly the required norm group. Hence, for every $n>0$,For $n=1$, the factor $1+2{\mathbb{Z}}_2$ is all of ${\mathbb{Z}}_2^{\times }$, consistently with ${\mathbb{Q}}_2({\xi }_2)={\mathbb{Q}}_2$.

Put ${\pi }_n={\zeta }_{p^n}-1$. The shifted cyclotomic polynomialis Eisenstein at $p$. Therefore it is irreducible, ${\pi }_n$ is a uniformizer, andso the extension is totally ramified. It is the splitting field of ${\Phi }_{p^n}$, and every automorphism is uniquelyThis proves the cyclotomic extension of a p-adic field isomorphismRestriction in the cyclotomic tower corresponds to reduction of $a$, so taking the inverse limit gives

The cyclotomic extension of a p-adic fieldhas degree $p-1$, is Galois with group $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$, and is totally ramified. Indeed, ${\zeta }_p-1$ is a root of the Eisenstein polynomial

Put $\alpha =\sqrt[p-1]{-p}$. Its polynomial $X^{p-1}+p$ is Eisenstein, so $L_2={\mathbb{Q}}_p(\alpha )$ is totally ramified of degree $p-1$. Every $(p-1)$st root of unity lies in ${\mathbb{Q}}_p$ by the Teichmuller lifts, so every root $\omega \alpha$ of this polynomial lies in $L_2$. Hence $L_2/{\mathbb{Q}}_p$ is also Galois.

For either $L_i/{\mathbb{Q}}_p$, the norm has every possible valuation because the residue-field degree is one. The norm units in a tamely totally ramified extension lie in the principal units $1+p{\mathbb{Z}}_p$: reduction of a unit norm is the $(p-1)$st power of its residue, hence is $1$. Part (b) says that this unit norm subgroup has index $p-1$, exactly the index of $1+p{\mathbb{Z}}_p$ in ${\mathbb{Z}}_p^{\times }$. ConsequentlyThe uniqueness clause in the existence theorem of local class field theory now gives