In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, the norm of an ideal is a generalization of a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of an element in the
field extension. It is particularly important in
number theory since it measures the size of an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
of a complicated
number ring in terms of an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
in a less complicated
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
. 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
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''K'' and
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
of ''B'' in a finite
separable extension ''L'' of ''K''. (this implies that ''B'' is also a Dedekind domain.) Let
and
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
:
is the unique
group homomorphism that satisfies
:
for all nonzero
prime ideals
of ''B'', where
is the
prime ideal of ''A'' lying below
.
Alternatively, for any
one can equivalently define
to be the
fractional ideal of ''A'' generated by the set
of
field norms of elements of ''B''.
For
, one has
, where