How Do Elements Really Factor in \(\mathbb {Z}[\sqrt{-5}]\)?

  • Scott T. ChapmanEmail author
  • Felix Gotti
  • Marly Gotti
Part of the Trends in Mathematics book series (TM)


Most undergraduate level abstract algebra texts use \(\mathbb {Z}[\sqrt{-5}]\) as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of how elements actually factor in this ring of algebraic integers. We offer here an interactive framework which shows that while \(\mathbb {Z}[\sqrt{-5}]\) is not a UFD, it does satisfy a slightly weaker factorization condition, known as half-factoriality. The arguments involved revolve around the Fundamental Theorem of Ideal Theory in algebraic number fields.

2010 AMS Mathematics Subject Classification

Primary 11R11 13F15 20M13 



It is a pleasure for the authors to thank the referee, whose helpful suggestions vastly improved the final version of this paper.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Sam Houston State UniversityHuntsvilleUSA
  2. 2.UC BerkeleyBerkeleyUSA
  3. 3.University of FloridaGainesvilleUSA

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