Basic Results
- Brill's Theorem: The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K.
- A prime p ramifies in K if, and only if, p divides ΔK .
- Stickelberger's Theorem:
- Minkowski's bound: Let n denote the degree of the extension K/Q and r2 the number of complex places of K, then
- Minkowski's Theorem: If K is not Q, then |ΔK| > 1 (this follows directly from the Minkowski bound).
- Hermite's Theorem: Let N be a positive integer. There are only finitely many algebraic number fields K with |ΔK| < N.
Read more about this topic: Discriminant Of An Algebraic Number Field
Famous quotes containing the words basic and/or results:
“There’s one basic rule you should remember about development charts that will save you countless hours of worry.... The fact that a child passes through a particular developmental stage is always more important than the age of that child when he or she does it. In the long run, it really doesn’t matter whether you learn to walk at ten months or fifteen months—as long as you learn how to walk.”
—Lawrence Kutner (20th century)
“There is ... in every child a painstaking teacher, so skilful that he obtains identical results in all children in all parts of the world. The only language men ever speak perfectly is the one they learn in babyhood, when no one can teach them anything!”
—Maria Montessori (1870–1952)