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Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e., $L= \mathbb{Q}(\sqrt[3]{\ell},\zeta_3)$.

We know that $\zeta_3 \in \mathsf{N}_{L/K}(L)$ where $\mathsf{N}_{L/K}$ denotes the norm from $L$ to $K$.

Q1. Is there a concrete way to check when $\zeta_3\in \mathsf{N}_{L/K}(E_L)$ where $E_L$ refers to the unit group of $L$?

Q2. For what proportion of primes of the form $1\pmod{9}$, can we say that $\zeta_3 \in \mathsf{N}_{L/K}(E_L)$?

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    $\begingroup$ Is "the unit group of $L$" just $L \setminus \{0\}$, or something else? $\endgroup$
    – LSpice
    Commented Apr 15 at 17:23
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    $\begingroup$ By unit group, I mean the the group of units of the ring of integers of $L$. $\endgroup$
    – debanjana
    Commented Apr 15 at 17:36
  • $\begingroup$ What's known about the structure of $E_L$? How hard is it to decide whether $\zeta_3$ is the norm of a unit, for a few small values of $\ell$? $\endgroup$
    – Gerry Myerson
    Commented Apr 16 at 3:12

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Of course you can check whether $\zeta_3$ is the norm of a unit by computing the unit group. There are a few publications about Scholz's unit knot (see Jehne's article https://eudml.org/doc/152174 ).

Thew second problem was just solved in the simpler case of when $-1$ is a norm of a unit in real quadratic number fields; see Stevenhagen's survey https://arxiv.org/abs/1806.06250 .

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