Abelian extension | EPFL Graph Search

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian. If a field K contains a primitive n-th root of unity and the n-

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

The valuation criterion for normal basis generators

If L/K is a finite Galois extension of local fields, then we say that the valuation criterion VC(L/K) holds if there is an integer d such that every element x is an element of L with valuation d generates a normal basis for L/K. Answering a question of Byott and Elder, we first prove that VC(L/K) holds if and only if the tamely ramified part of the extension L/K is trivial and every non-zero K[G]-submodule of L contains a unit of the valuation ring of L. Moreover, the integer d can take one value modulo [L:K] only, namely-d(L/K)-1, where d(L/K) is the valuation of the different of L/K. When K has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that VC(L/K) is valid for all extensions L/K in this context. When char K=0, we identify all abelian extensions L/K for which VC(L/K) is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.

2012

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