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:
“An absolute can only be given in an intuition, while all the rest has to do with analysis. We call intuition here the sympathy by which one is transported into the interior of an object in order to coincide with what there is unique and consequently inexpressible in it. Analysis, on the contrary, is the operation which reduces the object to elements already known.”
—Henri Bergson (18591941)
“As long as male behavior is taken to be the norm, there can be no serious questioning of male traits and behavior. A norm is by definition a standard for judging; it is not itself subject to judgment.”
—Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 1 (1991)