The euclidean algorithm for number fields and primitive roots

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Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.

Original languageEnglish (US)
Pages (from-to)181-190
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number1
StatePublished - 2012
Externally publishedYes


  • Euclidean algorithm
  • Large sieve
  • Primitive roots


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