## The cubics which are differences of two conjugates of an algebraic integer.(English)Zbl 1153.11336

Summary: We show that a cubic algebraic integer over a number field $$K$$, with zero trace is a difference of two conjugates over $$K$$ of an algebraic integer. We also prove that if $$N$$ is a normal cubic extension of the field of rational numbers, then every integer of $$N$$ with zero trace is a difference of two conjugates of an integer of $$N$$ if and only if the 3-adic valuation of the discriminant of $$N$$ is not 4.

### MSC:

 11R04 Algebraic numbers; rings of algebraic integers 11R16 Cubic and quartic extensions
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### References:

 [1] A. Dubickas, On numbers which are differences of two conjugates of an algebraic integer. Bull. Austral. Math. Soc. 65 (2002), 439-447. · Zbl 1028.11065 [2] A. Dubickas, C. J. Smyth, Variations on the theme of Hilbert’s Theorem 90. Glasg. Math. J. 44 (2002), 435-441. · Zbl 1112.11308 [3] S. Lang, Algebra. Addison-Wesley Publishing, Reading Mass. 1965. · Zbl 0193.34701 [4] A. Schinzel, Selected Topics on polynomials. University of Michigan, Ann Arbor, 1982. · Zbl 0487.12002 [5] T. Zaimi, On numbers which are differences of two conjugates over $$\mathbb{Q}$$of an algebraic integer. Bull. Austral. Math. Soc. 68 (2003), 233-242. · Zbl 1043.11073
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