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<p>The computation of the index ideal of a finite algebraic extension over a number field, is a deep task. Early, Hall gives a nice decomposition of the absolute index of cubic field (see Hall in Bull Am Math Soc 43(2):104–108, 1937). In this paper we prove a new version of the well-known structure Theorem of finitely generated modules over a PID which allows us to generalize the Hall’s result for cubic numbers to arbitrary extension of a fraction field of principal ideal domain. More precisely we show the existence of a monic triangular basis such as the sequence of its denominators is a factorial sequence. Further we express explicitly the ideal index as a product of a such factorial sequence and hence obtain an interesting decomposition of the ideal index over a PID. Some useful examples are also given.</p>

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On the decomposition of the ideal index over a PID

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • M. E. Charkani,
  • Omar Boughaleb

摘要

The computation of the index ideal of a finite algebraic extension over a number field, is a deep task. Early, Hall gives a nice decomposition of the absolute index of cubic field (see Hall in Bull Am Math Soc 43(2):104–108, 1937). In this paper we prove a new version of the well-known structure Theorem of finitely generated modules over a PID which allows us to generalize the Hall’s result for cubic numbers to arbitrary extension of a fraction field of principal ideal domain. More precisely we show the existence of a monic triangular basis such as the sequence of its denominators is a factorial sequence. Further we express explicitly the ideal index as a product of a such factorial sequence and hence obtain an interesting decomposition of the ideal index over a PID. Some useful examples are also given.