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]

See also

• Field norm
• Dedekind zeta function

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. 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