Nonsolvable polynomials with field discriminant 5a

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Abstract

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±pA for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL 2(5)5.10, and field discriminant 569. A closely related polynomial has degree 120, Galois group of the form SL 2(5)5.20, and field discriminant 5311. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 54.

Original languageEnglish (US)
Pages (from-to)289-322
Number of pages34
JournalInternational Journal of Number Theory
Volume7
Issue number2
DOIs
StatePublished - Mar 2011

Keywords

  • Nonsolvable
  • discriminant
  • number field
  • polynomial
  • ramification

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