On K-derived quartics

Andrew Bremner, Benjamin Carrillo

Research output: Contribution to journalArticlepeer-review


Let K be a number field. A K-derived polynomial f(x)∈K[x] is a polynomial that factors into linear factors over K, as do all of its derivatives. Such a polynomial is said to be proper if its roots are distinct. An unresolved question in the literature is whether or not there exists a proper Q-derived polynomial of degree 4. Some examples are known of proper K-derived quartics for a quadratic number field K, though other than Q(3), these fields have quite large discriminant. (The second known field is Q(3441).) The current paper describes a search for quadratic fields K over which there exist proper K-derived quartics. The search finds examples for K=Q(D) with D=…,−95,−41,−39,−19,21,31,89,… .

Original languageEnglish (US)
Pages (from-to)276-291
Number of pages16
JournalJournal of Number Theory
StatePublished - Nov 1 2016


  • Derived quartics
  • Elliptic curves
  • Linear factors
  • Quadratic fields

ASJC Scopus subject areas

  • Algebra and Number Theory


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