Ideal Norm
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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. Promi ...
, the norm of an ideal is a generalization of a norm of an element in the
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
. It is particularly important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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 In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
, Z, then the norm of a nonzero ideal ''I'' of a number ring ''R'' is simply the size of the finite
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
''R''/''I''.


Relative norm

Let ''A'' be a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
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'' ...
of ''B'' in a finite separable extension ''L'' of ''K''. (this implies that ''B'' is also a Dedekind domain.) Let \mathcal_A and \mathcal_B be the
ideal group In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral d ...
s of ''A'' and ''B'', respectively (i.e., the sets of nonzero
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
s.) Following the technique developed by
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...
, the norm map :N_\colon \mathcal_B \to \mathcal_A is the unique group homomorphism that satisfies :N_(\mathfrak q) = \mathfrak^ 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_B one can equivalently define N_(\mathfrak) to be the
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
of ''A'' generated by the set \ of field norms of elements of ''B''. For \mathfrak a \in \mathcal_A, one has N_(\mathfrak a B) = \mathfrak a^n, where n = : K/math>. The ideal norm of a principal ideal is thus compatible with the field norm of an element: :N_(xB) = N_(x)A. Let L/K be a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
of number fields with rings of integers \mathcal_K\subset \mathcal_L. Then the preceding applies with A = \mathcal_K, B = \mathcal_L, and for any \mathfrak b\in\mathcal_ we have :N_(\mathfrak b)= K \cap\prod_ \sigma (\mathfrak b), which is an element of \mathcal_. The notation N_ is sometimes shortened to N_, an
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors ...
that is compatible with also writing N_ for the field norm, as noted above. In the case K=\mathbb, it is reasonable to use positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s as the range for N_\, since \mathbb has trivial
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...
and unit group \, thus each nonzero
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
of \mathbb is generated by a uniquely determined positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
. Under this convention the relative norm from L down to K=\mathbb coincides with the absolute norm defined below.


Absolute norm

Let L be a number field with
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
\mathcal_L, and \mathfrak a a nonzero (integral) ideal of \mathcal_L. The absolute norm of \mathfrak a is :N(\mathfrak a) :=\left \mathcal_L: \mathfrak a\right \left, \mathcal_L/\mathfrak a\.\, 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_(a)\. The norm is completely multiplicative: if \mathfrak a and \mathfrak b are ideals of \mathcal_L, then :N(\mathfrak a\cdot\mathfrak b)=N(\mathfrak a)N(\mathfrak b). Thus the absolute norm extends uniquely to a group homomorphism :N\colon\mathcal_\to\mathbb_^\times, defined for all nonzero
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
s of \mathcal_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_(a)\\leq \left ( \frac\right )^s \sqrtN(\mathfrak a), where :* \Delta_L is the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
of L and :* s is the number of pairs of (non-real) complex
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
s of into \mathbb (the number of complex places of ).


See also

* Field norm * Dedekind zeta function


References

{{reflist Algebraic number theory Commutative algebra Ideals (ring theory)