Ideal norm

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In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.

Relative norm

Let A be a Dedekind domain with the field of fractions K and B be the integral closure of A in a finite separable extension L of K. (In particular, B is Dedekind then.) Let \operatorname {Id}(A) and \operatorname {Id}(B) be the ideal groups of A and B, respectively (i.e., the sets of fractional ideals.) Following (Serre 1979), the norm map

N_{{B/A}}:\operatorname {Id}(B)\to \operatorname {Id}(A)

is a homomorphism given by

N_{{B/A}}({\mathfrak q})={\mathfrak {p}}^{{[B/{\mathfrak q}:A/{\mathfrak p}]}},{\mathfrak q}\in \operatorname {Spec}B,{\mathfrak q}|{\mathfrak p}.

If L,K are local fields, N_{{B/A}}({\mathfrak {b}}) is defined to be a fractional ideal generated by the set \{N_{{L/K}}(x)|x\in {\mathfrak {b}}\}. This definition is equivalent to the above and is given in (Iwasawa 1986).

For {\mathfrak a}\in \operatorname {Id}(A), one has N_{{B/A}}({\mathfrak a}B)={\mathfrak a}^{n} where n=[L:K]. The definition is thus also compatible with norm of an element: N_{{B/A}}(xB)=N_{{L/K}}(x)A.[1]

Let L/K be a finite Galois extension of number fields with rings of integer {\mathcal {O}}_{K}\subset {\mathcal {O}}_{L}. Then the preceding applies with A={\mathcal {O}}_{K},B={\mathcal {O}}_{L} and one has

N_{{L/K}}(I)={\mathcal {O}}_{K}\cap \prod _{{\sigma \in G}}^{{}}\sigma (I)\,

which is an ideal of {\mathcal {O}}_{K}. The norm of a principal ideal generated by α is the ideal generated by the field norm of α.

The norm map is defined from the set of ideals of {\mathcal {O}}_{L} to the set of ideals of {\mathcal {O}}_{K}. It is reasonable to use integers as the range for N_{{L/{\mathbf {Q}}}}\, since Z has trivial ideal class group. This idea does not work in general since the class group may not be trivial.

Absolute norm

Let L be a number field with ring of integers {\mathcal {O}}_{L}, and \alpha a nonzero ideal of {\mathcal {O}}_{L}. Then the norm of \alpha is defined to be

N(\alpha )=\left[{\mathcal {O}}_{L}:\alpha \right]=|{\mathcal {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)|. For proof, cf. Marcus, theorem 22c, pp65ff.

The norm is also completely multiplicative in that if \alpha and \beta are ideals of {\mathcal {O}}_{L}, then N(\alpha \cdot \beta )=N(\alpha )N(\beta ). For proof, cf. Marcus, theorem 22a, pp65ff.

The norm of an ideal \alpha can be used to bound the norm of some nonzero element x\in \alpha by the inequality

|N(x)|\leq \left({\frac {2}{\pi }}\right)^{{r_{2}}}{\sqrt {|\Delta _{L}|}}N(\alpha )

where \Delta _{L} is the discriminant of L and r_{2} is the number of pairs of complex embeddings of L into {\mathbf {C}}.

See also

References

  1. ↑ Serre, 1. 5, Proposition 14.
  • Iwasawa, Kenkichi (1986), Local class field theory, Oxford Science Publications, New York: The Clarendon Press Oxford University Press, pp. viii+155, ISBN 0-19-504030-9, MR 863740 (88b:11080)  Unknown parameter |note= ignored (help)
  • Marcus, Daniel A. (1977), Number fields, New York: Springer-Verlag, pp. viii+279, ISBN 0-387-90279-1, MR 0457396 (56 #15601)  Unknown parameter |note= ignored (help)
  • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016)  Unknown parameter |note= ignored (help)
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