The <cyclotomic extension of a p-adic field>
$$
L_1=\mathbb Q_p(\zeta_p)
$$
has degree $p-1$, is Galois with group $(\mathbb Z/p\mathbb Z)^\times$, and is <totally ramified extension>[totally ramified]. Indeed, $\zeta_p-1$ is a root of the <Eisenstein polynomial>
$$
\frac{(X+1)^p-1}{X}.
$$

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 representative>[Teichmuller lifts], so every root $\omega\alpha$ of this polynomial lies in $L_2$. Hence $L_2/\mathbb Q_p$ is also <Galois extension>[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 $\mathbb Z_p^\times$. Consequently
$$
N_{L_1/\mathbb Q_p}(L_1^\times)
=p^{\mathbb Z}(1+p\mathbb Z_p)
=N_{L_2/\mathbb Q_p}(L_2^\times).
$$
The uniqueness clause in the <existence theorem of local class field theory> now gives
$$
L_1=L_2.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 136 / 1 / c / Solution

For a tamely totally ramified extension of degree $q-1$ of a local field with residue field $\mathbb F_q$, the unit norms are exactly the principal units. Reduction sends a unit norm to its $(q-1)$st power, and the <norm-unit index in an abelian local extension> shows that this containment has the correct index.

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 Incoming links: Norm units in a tamely totally ramified extension