Absolute Norm
Let be a number field with ring of integers, and a nonzero ideal of . Then the norm of is defined to be
By convention, the norm of the zero ideal is taken to be zero.
If is a principal ideal with, then . For proof, cf. Marcus, theorem 22c, pp65ff.
The norm is also completely multiplicative in that if and are ideals of, then . For proof, cf. Marcus, theorem 22a, pp65ff.
The norm of an ideal can be used to bound the norm of some nonzero element by the inequality
where is the discriminant of and is the number of pairs of complex embeddings of into .
Read more about this topic: Ideal Norm
Famous quotes containing the words absolute and/or norm:
“The appalling thing is the degree of charity women are capable of. You see it all the time ... love lavished on absolute fools. Love’s a charity ward, you know.”
—Lawrence Durrell (1912–1990)
“To be told that our child’s behavior is “normal” offers little solace when our feelings are badly hurt, or when we worry that his actions are harmful at the moment or may be injurious to his future. It does not help me as a parent nor lessen my worries when my child drives carelessly, even dangerously, if I am told that this is “normal” behavior for children of his age. I’d much prefer him to deviate from the norm and be a cautious driver!”
—Bruno Bettelheim (20th century)