Dual basis in a field extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite field 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 is true if L is separable over K; it is always true if K is a perfect field, including when K is finite or of characteristic zero.

A dual basis is not a specific basis like the polynomial basis or the normal basis; rather for any given basis, it is another associated basis which is useful for computations.

We say that two bases of a finite field ${\displaystyle L={\text{GF}}(p^{n})}$ over ${\displaystyle K={\text{GF}}(p)=\mathbb {Z} /p\mathbb {Z} }$,

${\displaystyle B_{1}=\{\alpha _{0},\alpha _{1},\ldots ,\alpha _{n-1}\},\qquad B_{2}=\{\beta _{0},\beta _{1},\ldots ,\beta _{n-1}\},}$

are dual to each other provided

${\displaystyle \operatorname {Tr} (\alpha _{i}\cdot \beta _{j})={\begin{cases}1&{\text{if }}\ i=j\\0&{\text{if }}\ i\neq j.\end{cases}}}$

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

${\displaystyle \operatorname {Tr} (\gamma )=\sum _{i=0}^{n-1}\gamma ^{p^{i}}}$

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).