On the irreducibility of Hecke polynomials

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Let Tn, k(X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials Tn, k(X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that Tn, k(X) is irreducible and has the full symmetric group as Galois group, then the same is true of Tp, k(X) for each prime p le; 4,000,000.

Original languageEnglish (US)
Pages (from-to)1725-1731
Number of pages7
JournalMathematics of Computation
Issue number263
StatePublished - Jul 2008

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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