Normal basis | EPFL Graph Search

In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. Let be a Galois extension with Galois group . The classical normal basis theorem states that there is an element such that forms a basis of K, considered as a vector space over F. That is, any element can be written uniquely as for some elements A normal basis contrasts with a primitive element basis of the form , where is an element whose minimal polynomial has degree . A field extension K / F with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules is of form for some . Since is a linear basis of F[G] over F, it follows easily that is bijective iff generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if K / F is finite Galois extension, then as left -module. In terms of representations of G over F, this means that K is isomorphic to the regular representation. For finite fields this can be stated as follows: Let denote the field of q elements, where q = pm is a prime power, and let denote its extension field of degree n ≥ 1. Here the Galois group is with a cyclic group generated by the q-power Frobenius automorphism with Then there exists an element β ∈ K such that is a basis of K over F. In case the Galois group is cyclic as above, generated by with the normal basis theorem follows from two basic facts.

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

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