Editing Dual basis in a field extension (section)

{{No inline|date=May 2025}}
In [[mathematics]], the [[linear algebra]] concept of [[dual basis]] can be applied in the context of a [[finite extension|finite]] [[field extension]] ''L''/''K'', by using the [[field trace]]. This requires the property that the field trace ''Tr''<sub>''L''/''K''</sub>  provides a [[non-degenerate]] [[quadratic form]] over ''K''. This can be guaranteed if the extension is [[separable extension|separable]]; it is automatically true if ''K'' is a [[perfect field]], and hence in the cases where ''K'' is [[finite field|finite]], or of [[characteristic (algebra)|characteristic]] zero.

A '''dual basis''' () is not a concrete [[basis (linear algebra)|basis]] like the [[polynomial basis]] or the [[normal basis]]; rather it provides a way of using a second basis for computations.

Consider two bases for elements in a finite field, GF(''p''<sup>''m''</sup>):

:<math>B_1 = {\alpha_0, \alpha_1, \ldots, \alpha_{m-1}}</math>
and
:<math>B_2 = {\gamma_0, \gamma_1, \ldots, \gamma_{m-1}}</math>

then ''B''<sub>2</sub> can be considered a dual basis of ''B''<sub>1</sub> provided

:<math>\operatorname{Tr}(\alpha_i\cdot \gamma_j) = \begin{cases}
0, & \operatorname{if}\ i \neq j
\\ 1, & \operatorname{otherwise}.
\end{cases}</math>

Here the [[field trace|trace]] of a value in GF(''p''<sup>''m''</sup>) can be calculated as follows:

:<math>\operatorname{Tr}(\beta ) = \sum_{i=0}^{m-1} \beta^{p^i}</math>

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 basis|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).