Galois number fields with small root discriminant

John Jones, David P. Roberts

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 A5 2 . 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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Galois group
Discriminant
Number field
Galois field
Roots

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Galois number fields with small root discriminant. / Jones, John; Roberts, David P.

In: Journal of Number Theory, Vol. 122, No. 2, 02.2007, p. 379-407.

Research output: Contribution to journalArticle

Jones, John ; Roberts, David P. / Galois number fields with small root discriminant. In: Journal of Number Theory. 2007 ; Vol. 122, No. 2. pp. 379-407.
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