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
VL - 168
SP - 276
EP - 291
JO - Journal of Number Theory
JF - Journal of Number Theory
SN - 0022-314X
ER -