### Abstract

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 language | English (US) |
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Pages (from-to) | 276-291 |

Number of pages | 16 |

Journal | Journal of Number Theory |

Volume | 168 |

DOIs | |

State | Published - Nov 1 2016 |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*168*, 276-291. https://doi.org/10.1016/j.jnt.2016.04.024