whose entries are the trace of over all pairs , is called the discriminant of the basis . The ideal in generated by all discriminants of the form
is called the discriminant ideal of over , and denoted .
2 Alternative notations
The discriminant is sometimes denoted with instead of . In the context of number fields, one often writes for where and are the rings of algebraic integers of and . If or is omitted, it is typically assumed to be or .
The discriminant is so named because it allows one to determine which ideals of are ramified in . Specifically, the prime ideals of that ramify in are precisely the ones that contain the discriminant ideal . In the case , a theorem of Minkowski (http://planetmath.org/MinkowskisConstant) that any ring of integers of a number field larger than has discriminant strictly smaller than itself, and this fact combined with the previous result shows that any number field admits at least one ramified prime over .
4 Other types of discriminants
In the special case where is a primitive separable field extension of degree , the discriminant is equal to the polynomial discriminant (http://planetmath.org/PolynomialDiscriminant) of the minimal polynomial of over .
The discriminant of an elliptic curve can be obtained by taking the polynomial discrimiant of its Weierstrass polynomial, and the modular discriminant of a complex lattice equals the discriminant of the elliptic curve represented by the corresponding lattice quotient.
|Date of creation||2013-03-22 12:37:57|
|Last modified on||2013-03-22 12:37:57|
|Last modified by||djao (24)|