- Norm of an ideal
The norm of an ideal is defined in
algebraic number theory . Let Ksubset L be two number fields with rings of integers O_Ksubset O_L. Suppose that the extension L/K is aGalois extension with:G= extstyle{Gal}(L/K).
The norm of an ideal I of O_L is defined as follows
:N_K^L(I)=O_K capprod_{sigma in G}^{} sigma (I)
which is an ideal of O_K. The norm of a
principal ideal generated by "α" is the ideal generated by thefield norm of "α".The norm map is defined from the set of ideals of O_L. to the set of ideals of O_K. It is reasonable to use integers as the range for the norm map
:N_mathbb{Q}^L(I)
since Z is a
principal ideal domain . This idea doesn't work in general since class group is usually non-trivial.Alternate Formulation
Let L be a number field with ring of integers O_L, and alpha a nonzero ideal of O_L. Then the norm of alpha is defined to be :N(alpha) =left [ O_L: alpha ight ] =|O_L/alpha|.By convention, the norm of the zero ideal is taken to be zero.
If alpha is a principal ideal with alpha=(a), then N(alpha)=|N(a)|.
The norm is also completely multiplicative in that if alpha and eta are ideals of O_L, then N(alpha*eta)=N(alpha)N(eta).
The norm of an ideal alpha can be used to bound the norm of some nonzero element xin alpha by the inequality:N(x)|leq left ( frac{2}{pi} ight ) ^ {r_2} sqrtN(alpha)where Delta_L is the discriminant of L and r_2 is the number of pairs of complex embeddings of L into mathbb{C}.
ee also
*
Dedekind zeta function References
* Daniel A. Marcus, "Number Fields", third edition, Springer-Verlag, 1977
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