In
algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the
ring of integers of an
algebraic number field ''K'', with respect to the
field trace
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a field and ''L'' a finite extension (and hence a ...
. It then encodes the
ramification data for
prime ideals of the ring of integers. It was introduced by
Richard Dedekind in 1882.
Definition
If ''O''
''K'' is the ring of integers of ''K'', and ''tr'' denotes the field trace from ''K'' to the
rational number field
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 ...
Q, then
:
is an
integral quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
on ''O''
''K''. Its
discriminant as quadratic form need not be +1 (in fact this happens only for the case ''K'' = Q). Define the ''inverse different'' or ''codifferent''
or ''Dedekind's complementary module''
as the set ''I'' of ''x'' ∈ ''K'' such that tr(''xy'') is an integer for all ''y'' in ''O''
''K'', then ''I'' is a
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 ...
of ''K'' containing ''O''
''K''. By definition, the different ideal δ
''K'' is the inverse fractional ideal ''I''
−1: it is an ideal of ''O''
''K''.
The
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 ...
of ''δ''
''K'' is equal to the ideal of ''Z'' generated by the
field discriminant ''D''
''K'' of ''K''.
The ''different of an element'' α of ''K'' with minimal polynomial ''f'' is defined to be δ(α) = ''f''′(α) if α generates the field ''K'' (and zero otherwise):
[ we may write
:
where the α(''i'') run over all the roots of the characteristic polynomial of α other than α itself.] The different ideal is generated by the differents of all integers α in ''O''''K''. This is Dedekind's original definition.
The different is also defined for a finite degree extension of local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
s. It plays a basic role in Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
for p-adic field
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
s.
Relative different
The relative different δ''L'' / ''K'' is defined in a similar manner for an extension of number fields ''L'' / ''K''. The relative norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Formal definition
Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of '' ...
of the relative different is then equal to the relative discriminant Δ''L'' / ''K''. In a tower of fields
In mathematics, a tower of fields is a sequence of field extensions
:
The name comes from such sequences often being written in the form
:\begin\vdots \\ , \\ F_2 \\ , \\ F_1 \\ , \\ \ F_0. \end
A tower of fields may be finite or infinite.
Exam ...
''L'' / ''K'' / ''F'' the relative differents are related by δ''L'' / ''F'' = δ''L'' / ''K''''δ''''K'' / ''F''.
The relative different equals the annihilator of the relative Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebr ...
module :
The ideal class of the relative different δ''L'' / ''K'' is always a square in the 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 mea ...
of ''O''''L'', the ring of integers of ''L''. Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of ''O''''K'': indeed, it is the square of the Steinitz class for ''O''''L'' as a ''O''''K''-module.
Ramification
The relative different encodes the ramification data of the field extension ''L'' / ''K''. A prime ideal ''p'' of ''K'' ramifies in ''L'' if the factorisation of ''p'' in ''L'' contains a prime of ''L'' to a power higher than 1: this occurs if and only if ''p'' divides the relative discriminant Δ''L'' / ''K''. More precisely, if
:''p'' = ''P''1''e''(1) ... ''P''''k''''e''(''k'')
is the factorisation of ''p'' into prime ideals of ''L'' then ''P''''i'' divides the relative different δ''L'' / ''K'' if and only if ''P''''i'' is ramified, that is, if and only if the ramification index ''e''(''i'') is greater than 1. The precise exponent to which a ramified prime ''P'' divides δ is termed the differential exponent of P and is equal to ''e'' − 1 if ''P'' is tamely ramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
: that is, when ''P'' does not divide ''e''. In the case when ''P'' is wildly ramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
the differential exponent lies in the range ''e'' to ''e'' + ''e''ν''P''(e) − 1. The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[, p. 115]
Local computation
The different may be defined for an extension of local fields ''L'' / ''K''. In this case we may take the extension to be simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, generated by a primitive element α which also generates a power integral basis In mathematics, a monogenic field is an algebraic number field ''K'' for which there exists an element ''a'' such that the ring of integers ''O'K'' is the subring Z 'a''of ''K'' generated by ''a''. Then ''O'K'' is a quotient of the polynomi ...
. If ''f'' is the minimal polynomial for α then the different is generated by ''f(α).
Notes
References
*
* . Retrieved 5 August 2009
*
*
*
*
*
* {{citation , last=Weiss , first=Edwin , title=Algebraic Number Theory , publisher=Chelsea Publishing
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
, edition=2nd unaltered , year=1976 , isbn=0-8284-0293-0 , zbl=0348.12101 , url-access=registration , url=https://archive.org/details/algebraicnumbert00weis_0
Algebraic number theory