= Effective norm-form height estimate
Let $L$ be a fixed <number field> and let $V\subset L$ be a fixed rational linear subspace for which the associated <norm form> has no unit-family degeneracy. There are effective constants $A,C>0$ such that every $z\in V\cap\mathcal O_L$ with nonzero <field norm> $m$ satisfies
$$
H(z)\leq C|m|^A.
$$
To prove this, factor the principal ideal $(z)$, write its generators as a bounded factor times powers of fundamental units, use the linear relations defining $V$, and apply the <Baker lower bound for a homogeneous linear form in logarithms> to bound the unit exponents by $O(\log|m|)$. For $L=\mathbb Q(\alpha)K$ with $[\mathbb Q(\alpha):\mathbb Q]=3$, $[K:\mathbb Q]=2$, and $V=\operatorname{span}_{\mathbb Q}\{1,\alpha,\omega\}$ for a basis $1,\omega$ of $K$, the coprime degrees rule out the degeneracy.
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