Ideal norm

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 field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let ${\mathcal {I}}_{A}$ and ${\mathcal {I}}_{B}$ be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map

N_{B/A}\colon {\mathcal {I}}_{B}\to {\mathcal {I}}_{A}

is the unique group homomorphism that satisfies

N_{B/A}({\mathfrak {q}})={\mathfrak {p}}^{[B/{\mathfrak {q}}:A/{\mathfrak {p}}]}

for all nonzero prime ideals ${\mathfrak {q}}$ of B, where ${\mathfrak {p}}={\mathfrak {q}}\cap A$ is the prime ideal of A lying below ${\mathfrak {q}}$ .

Alternatively, for any ${\mathfrak {b}}\in {\mathcal {I}}_{B}$ one can equivalently define $N_{B/A}({\mathfrak {b}})$ to be the fractional ideal of A generated by the set $\{N_{L/K}(x)|x\in {\mathfrak {b}}\}$ of field norms of elements of B.^[1]

For ${\mathfrak {a}}\in {\mathcal {I}}_{A}$ , one has $N_{B/A}({\mathfrak {a}}B)={\mathfrak {a}}^{n}$ , where $n=[L:K]$ . The ideal norm of a principal ideal is thus compatible with the field norm of an element: $N_{B/A}(xB)=N_{L/K}(x)A.$ ^[2]

Let $L/K$ be a Galois extension of number fields with rings of integers ${\mathcal {O}}_{K}\subset {\mathcal {O}}_{L}$ . Then the preceding applies with $A={\mathcal {O}}_{K},B={\mathcal {O}}_{L}$ , and for any ${\mathfrak {b}}\in {\mathcal {I}}_{{\mathcal {O}}_{L}}$ we have

N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}({\mathfrak {b}})=K\cap \prod _{\sigma \in \operatorname {Gal} (L/K)}\sigma ({\mathfrak {b}}),

which is an element of ${\mathcal {I}}_{{\mathcal {O}}_{K}}$ . The notation $N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}$ is sometimes shortened to $N_{L/K}$ , an abuse of notation that is compatible with also writing $N_{L/K}$ for the field norm, as noted above.

In the case $K=\mathbb {Q}$ , it is reasonable to use positive rational numbers as the range for $N_{{\mathcal {O}}_{L}/\mathbb {Z} }\,$ since $\mathbb {Z}$ has trivial ideal class group and unit group $\{\pm 1\}$ , thus each nonzero fractional ideal of $\mathbb {Z}$ is generated by a uniquely determined positive rational number. Under this convention the relative norm from $L$ down to $K=\mathbb {Q}$ coincides with the absolute norm defined below.

Absolute norm

Let $L$ be a number field with ring of integers ${\mathcal {O}}_{L}$ , and ${\mathfrak {a}}$ a nonzero (integral) ideal of ${\mathcal {O}}_{L}$ .

The absolute norm of ${\mathfrak {a}}$ is

N({\mathfrak {a}}):=\left[{\mathcal {O}}_{L}:{\mathfrak {a}}\right]=\left|{\mathcal {O}}_{L}/{\mathfrak {a}}\right|.\,

By convention, the norm of the zero ideal is taken to be zero.

If ${\mathfrak {a}}=(a)$ is a principal ideal, then

N({\mathfrak {a}})=\left|N_{L/\mathbb {Q} }(a)\right|

.^[3]

The norm is completely multiplicative: if ${\mathfrak {a}}$ and ${\mathfrak {b}}$ are ideals of ${\mathcal {O}}_{L}$ , then

N({\mathfrak {a}}\cdot {\mathfrak {b}})=N({\mathfrak {a}})N({\mathfrak {b}})

.^[3]

Thus the absolute norm extends uniquely to a group homomorphism

N\colon {\mathcal {I}}_{{\mathcal {O}}_{L}}\to \mathbb {Q} _{>0}^{\times },

defined for all nonzero fractional ideals of ${\mathcal {O}}_{L}$ .

The norm of an ideal ${\mathfrak {a}}$ can be used to give an upper bound on the field norm of the smallest nonzero element it contains: there always exists a nonzero $a\in {\mathfrak {a}}$ for which

\left|N_{L/\mathbb {Q} }(a)\right|\leq \left({\frac {2}{\pi }}\right)^{s}{\sqrt {\left|\Delta _{L}\right|}}N({\mathfrak {a}}),

where $\Delta _{L}$ is the discriminant of $L$ and $s$ is the number of pairs of (non-real) complex embeddings of $L$ into $\mathbb {C}$ (the number of complex places of $L$ ).^[4]

References

^ Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545
^ Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin Jay, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 0554237
^ ^a ^b Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396
^ Neukirch, Jürgen (1999), Algebraic number theory, Berlin: Springer-Verlag, Lemma 6.2, doi:10.1007/978-3-662-03983-0, ISBN 3-540-65399-6, MR 1697859

[Janusz-1] Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545

[Serre-2] Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin Jay, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 0554237

[Marcus-3] Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396

[Neukirch-4] Neukirch, Jürgen (1999), Algebraic number theory, Berlin: Springer-Verlag, Lemma 6.2, doi:10.1007/978-3-662-03983-0, ISBN 3-540-65399-6, MR 1697859

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[2]

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[4]

Relative norm

Absolute norm

See also

References