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2017 Miklós Schweitzer, 4
Tags: number theory , number fields
Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which $$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.