Galois number fields with small root discriminant

John W. Jones, David P. Roberts

    Research output: Contribution to journalArticlepeer-review

    11 Scopus citations

    Abstract

    We pose the problem of identifying the set K (G, Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A4, A5, A6 and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL3 (2), A7, S7, PGL2 (7), SL2 (8), Σ L2 (8), PGL2 (9), P Γ L2 (9), PSL2 (11), and A52 . 2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K (G, Ω) is empty.

    Original languageEnglish (US)
    Pages (from-to)379-407
    Number of pages29
    JournalJournal of Number Theory
    Volume122
    Issue number2
    DOIs
    StatePublished - Feb 2007

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