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
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