MathDB

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

\{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.

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