## On numbers which are differences of two conjugates over $$\mathbb Q$$ of an algebraic integer.(English)Zbl 1043.11073

Let $$L$$ be a number field and let $$K$$ be its subfield. Hilbert’s Theorem 90 implies that if $$L/K$$ is cyclic then $$\beta \in L$$ can be written as $$\alpha-\alpha'$$, where $$\alpha, \alpha' \in L$$ are $$K$$-conjugate if and only if the $$L/K$$-trace of $$\beta$$ is equal to zero. Recently, the reviewer and C. J. Smyth considered several variations of this problem. In particular, the reviewer obtained partial results in the following problem: describe all algebraic integers $$\beta$$, which can be written as $$\alpha-\alpha'$$, where $$\alpha, \alpha'$$ are $$K$$-conjugate algebraic integers.
In this paper, the author obtains several results for cubic algebraic integers over $$\mathbb Q$$. Let $$x^3+px+q$$, where $$p,q\in\mathbb Z$$, be the minimal polynomial of $$\beta$$, and let $$N$$ be the normal closure of $$\mathbb Q(\beta)$$ over $$\mathbb Q$$. The author proves that then $$\beta$$ is a difference of two conjugates of an algebraic integer of degree at most 3 over the field $$N$$. He also describes all situations when such $$\beta$$ is a difference of two conjugates (this time over $$\mathbb Q$$) of an algebraic integer lying in $$N$$. In particular, he proves that the latter holds if the highest power of 3 dividing the discriminant of $$x^3+px+q$$ is different from 3, 4 and 5. The most difficult situation occurs when this highest power is 3, in which case some additional conditions for the representation of $$\beta$$ by a difference must hold.

### MSC:

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

### Keywords:

Hilbert’s Theorem 90; cubic extensions
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### References:

 [1] Schinzel, Selected topics on polynomials (1982) [2] Dubickas, Bull. Austral. Math. Soc. 65 pp 439– (2002) [3] Lang, Algebra (1965)
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