TY - JOUR

T1 - On K-derived quartics

AU - Bremner, Andrew

AU - Carrillo, Benjamin

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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,… .

AB - 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,… .

KW - Derived quartics

KW - Elliptic curves

KW - Linear factors

KW - Quadratic fields

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U2 - 10.1016/j.jnt.2016.04.024

DO - 10.1016/j.jnt.2016.04.024

M3 - Article

AN - SCOPUS:84976582719

SN - 0022-314X

VL - 168

SP - 276

EP - 291

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -