Further Properties
The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the minimal polynomial.
In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is an ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in OK/I – i.e. the cardinality of this finite ring. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.
Read more about this topic: Field Norm
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)