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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
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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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