Fractional ideal | EPFL Graph Search

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 domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity. Definition and basic results Let R be an integral domain, and let K = \operatorname{Frac}R be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r \in R such that rI\subseteq R. The element r can be thought of as clearing out the denominators in I, hence the name fractional ideal. The principal fractional ideals are those R-submodules of

Réseaux idéaux sur des corps CM et des corps totalement réels

Ivan Suarez Atias

This thesis deals with the study of ideal lattices over number fields. Let K be a number field, which is assumed to be CM or totally real. An ideal lattice over K is a pair (I,b), where I is a fractional ideal of K and b : I × I → R is a symmetric positive definite bilinear form such that b(x, y) = Tr(αxy) for a totally positive α ∈ K ⊗ R. The ideal lattice defined previously will be denoted by (I,α). In the first part, we will focus on constructing modular lattices over number fields. In particular the case of cyclotomic fields will be treated more carefully. A modular lattice is a lattice which is similar to its dual lattice. H.-G. Quebbemann introduced this notion in 1995 and in 1997, and he also defined a notion of analytic extremality for these lattices. It seemed promising to look for ideal lattices which were modular. This investigation led to the introduction of Arakelov modular lattices. This notion has turned out to be very interesting. If K = Q(ζn ) is a cvclotomic field, then the set of levels ℓ for which there exists an Arakelov modular lattice of level ℓ over Q(ζn ) is explicitly given in this thesis. Moreover, assume that K is a CM field, and that we are given an Arakelov modular lattice (I,α) of level ℓ over K. Under an assumption on K (which is satisfied for cyclotomic fields), we describe a way to compute all the Arakelov modular lattices of level ℓ over K. In the second part of this thesis, a construction is introduced which enables us to associate a totally real field K' to a given CM-field K. This field K' has the following property : if (I,α) is an ideal lattice over K. where I is an ambiguous ideal, then there exists an ideal lattice over K' isometric (as a lattice) to (I,α). This leads to the construction of several classic lattices over totally real fields. It also enables us to bound the Euclidean minimum of the field K' with respect to that of the field K.

EPFL2005

Construction Of Self-Dual Integral Normal Bases In Abelian Extensions Of Finite And Local Fields

Erik Jarl Pickett

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010

Réseaux idéaux sur des corps CM et des corps totalement réels

Ivan Suarez Atias

EPFL2005

Construction Of Self-Dual Integral Normal Bases In Abelian Extensions Of Finite And Local Fields

Erik Jarl Pickett

Let

F/E

be a finite Galois extension of fields with abelian Galois group

\Gamma

. A self-dual normal basis for

F/E

is a normal basis with the additional property that

Tr_{F/E}(g(x),h(x))=\delta_{g,h}

for

g,h\in\Gamma

. Bayer-Fluckiger and Lenstra have shown that when

char(E)\neq 2

, then

F

admits a self-dual normal basis if and only if

[F:E]

is odd. If

F/E

is an extension of finite fields and

char(E)=2

, then

F

admits a self-dual normal basis if and only if the exponent of

\Gamma

is not divisible by

4

. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let

K

be a finite extension of

\Q_p

, let

L/K

be a finite abelian Galois extension of odd degree and let

\bo_L

be the valuation ring of

L

. We define

A_{L/K}

to be the unique fractional

\bo_L

-ideal with square equal to the inverse different of

L/K

. It is known that a self-dual integral normal basis exists for

A_{L/K}

if and only if

L/K

is weakly ramified. Assuming

p\neq 2

, we construct such bases whenever they exist.

2010