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Some results on integrally closed domains. (English) Zbl 1428.12009

Dani, Shrikrishna G. (ed.) et al., Contributions in algebra and algebraic geometry. International conference on algebra, discrete mathematics and applications, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra, India, December 9–11, 2017. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 738, 75-80 (2019).
Summary: Let \(K_1\), \(K_2\) be finite separable extensions of a field \(K\) which are linearly disjoint over \(K\). In 2017, we proved that if the integral closures \(S_1\), \(S_2\) of a valuation ring \(R_v\) of \(K\) with perfect residue field, in \(K_1\), \(K_2\) respectively are free \(R_v\)-modules and the composite ring \(S_1S_2\) is integrally closed, then the discriminant of either \(S_1/R_v\) or \(S_2/R_v\) is the unit ideal (cf. [J. Pure Appl. Algebra 222, No. 11, 3560–3565 (2018; Zbl 1417.12003)]). From this result it was deduced that in particular when \(K_1\), \(K_2\) are algebraic number fields with the composite ring \(A_{K_1}A_{K_2}\) integrally closed, \(A_{K_i}\) being the ring of algebraic integers of \(K_i\), then the relative discriminants of \(K_1/K\) and \(K_2/K\) are coprime. In this paper, we show that the converse of both the above results holds.
For the entire collection see [Zbl 1428.13001].

MSC:

12J10 Valued fields
11R04 Algebraic numbers; rings of algebraic integers

Citations:

Zbl 1417.12003
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References:

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