## 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

### Keywords:

algebraic number; algebraic integer; conjugate

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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