# 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 ${\displaystyle {\mathcal {I}}_{A}}$  and ${\displaystyle {\mathcal {I}}_{B}}$  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

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

is the unique group homomorphism that satisfies

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

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

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

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

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

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

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

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

## Absolute norm

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

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

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

If ${\displaystyle {\mathfrak {a}}=(a)}$  is a principal ideal, then ${\displaystyle N({\mathfrak {a}})=\left|N_{L/\mathbb {Q} }(a)\right|}$ .[3]

The norm is completely multiplicative: if ${\displaystyle {\mathfrak {a}}}$  and ${\displaystyle {\mathfrak {b}}}$  are ideals of ${\displaystyle {\mathcal {O}}_{L}}$ , then ${\displaystyle N({\mathfrak {a}}\cdot {\mathfrak {b}})=N({\mathfrak {a}})N({\mathfrak {b}})}$ .[3] Thus the absolute norm extends uniquely to a group homomorphism

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

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

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

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

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

## References

1. ^ Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, 7 (second ed.), Providence, Rhode Island: American Mathematical Society, Proposition I.8.2, ISBN 0-8218-0429-4, MR 1362545
2. ^ Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, 67, translated by Greenberg, Marvin Jay, New York: Springer-Verlag, 1.5, Proposition 14, ISBN 0-387-90424-7, MR 0554237
3. ^ a b Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, Theorem 22c, ISBN 0-387-90279-1, MR 0457396
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