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 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.

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.

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

:<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 B₂ can be considered a dual basis of B₁ 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 trace of a value in GF(p^m) 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 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).

References

• , Definition 2.30, p. 54.