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Linear relations of four conjugates of an algebraic number
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 04 September 2025, pp. 1-16
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We characterize all algebraic numbers
$\alpha $ of degree 
$d\in \{4,5,6,7\}$ for which there exist four distinct algebraic conjugates 
$\alpha _1$, 
$\alpha _2$, 
$\alpha _3$, and 
$\alpha _4$ of 
$\alpha $ satisfying the relation 
$\alpha _{1}+\alpha _{2}=\alpha _{3}+\alpha _{4}$. In particular, we prove that an algebraic number 
$\alpha $ of degree 6 satisfies this relation with 
$\alpha _{1}+\alpha _{2}\notin \mathbb {Q}$ if and only if 
$\alpha $ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of 
$\mathbb {Q}(\alpha )$ for such algebraic numbers 
$\alpha $. We also consider similar relations 
$\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}=0$ and 
$\alpha _{1}+\alpha _{2}+\alpha _{3}=\alpha _{4}$ for algebraic numbers of degree up to 7.
