Dual basis in a field extension

Last updated March 25, 2019

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace Tr_L/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Linear algebra is the branch of mathematics concerning linear equations such as

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I, its dual set is a set B^∗ of vectors in the dual space V^∗ with the same index set I such that B and B^∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V^∗. If it does span V^∗, then B^∗ is called the dual basis or reciprocal basis for the basis B.

A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations.

In mathematics, a set $B$ of elements (vectors) in a vector space $V$ is called a basis, if every element of $V$ may be written in a unique way as a (finite) linear combination of elements of $B$ . The coefficients of this linear combination are referred to as components or coordinates on $B$ of the vector. The elements of a basis are called basis vectors.

In mathematics, a polynomial basis is a basis of a polynomial ring, viewed as a vector space over the field of coefficients, or as a free module over the ring of coefficients. The most common polynomial basis is the monomial basis consisting of all monomials. Other useful polynomial bases are the Bernstein basis and the various sequences of orthogonal polynomials.

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.

Consider two bases for elements in a finite field, GF(p^m):

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod $p$ when $p$ is a prime number.

B_{1}={\alpha _{0},\alpha _{1},\ldots ,\alpha _{m-1}}

and

B_{2}={\gamma _{0},\gamma _{1},\ldots ,\gamma _{m-1}}

then B₂ can be considered a dual basis of B₁ provided

\operatorname {Tr} (\alpha _{i}\cdot \gamma _{j})=\left\{{\begin{matrix}0,&\operatorname {if} \ i\neq j\\1,&\operatorname {otherwise} \end{matrix}}\right.

Here the trace of a value in GF(p^m) can be calculated as follows:

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.

\operatorname {Tr} (\beta )=\sum _{i=0}^{m-1}\beta ^{p^{i}}

Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity (usually 1).

In linear algebra, a basis for a vector space of dimension n is a set of n vectors (α₁, …, α_n), called basis vectors, with the property that every vector in the space can be expressed as a unique linear combination of the basis vectors. The matrix representations of operators are also determined by the chosen basis. Since it is often desirable to work with more than one basis for a vector space, it is of fundamental importance in linear algebra to be able to easily transform coordinate-wise representations of vectors and operators taken with respect to one basis to their equivalent representations with respect to another basis. Such a transformation is called a change of basis.

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