PSL septimic fields with a power basis
We give an infinite set of distinct monogenic septimic fields whose normal closure has Galois group .
We give an infinite set of distinct monogenic septimic fields whose normal closure has Galois group .
A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K.
A commutative order in a quaternion algebra is called selective if it embeds into some, but not all, of the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with type number 3 or larger. The proof extends to a few other closely related orders.
We prove density modulo of the sets of the formwhere is a pair of rationally independent algebraic integers of degree satisfying some additional assumptions, and is any sequence of real numbers.
Pour tout , on calcule un rang tel que tout entier algébrique de degré au moins ait deux conjugués vérifiant . De plus, on donne une nouvelle preuve de l’égalité .