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On numbers which are differences of two conjugates of an algebraic integer. (English) Zbl 1028.11065

The author investigates which algebraic integers \(\beta\) can be written as the difference \(\alpha - \alpha'\) of two algebraic integers conjugate over a given number field \(K\). The analogous question under the assumption that \(\beta, \alpha,\alpha'\) are conjugate algebraic numbers but not necessarily integers was solved by the author and C. J. Smyth [Glasg. Math. J. 44, 435-441 (2002; Zbl 1112.11308)]. He shows that if \(\beta\) is of degree \(2\) or \(3\) over \(K\) then a necessary and sufficient condition for \(\beta = \alpha-\alpha'\) is that the trace of \(\beta\) over \(K\) is \(0\). He also shows that if the minimal polynomial of \(\beta\) over \(K\) is of the form \(f(x^e)\) with \(f \in \mathbb Z_K[x]\) then \(\beta = \alpha-\alpha'\). He considers also the analogous question when “difference” is replaced by “quotient” and the condition on \(\alpha\) is that it be a unit.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions
11R21 Other number fields
11R32 Galois theory

Citations:

Zbl 1112.11308
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References:

[1] Smyth, Acta Arith. 40 pp 333– (1982)
[2] Hilbert, Jahresber. Deutsch. Math.-Verein 4 pp 175– (1897)
[3] Lang, Algebra (1965)
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