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:
“Theres a basic rule which runs through all kinds of music, kind of an unwritten rule. I dont know what it is. But Ive got it.”
—Ron Wood (b. 1947)
“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 (18701952)