Dual basis in a field extension

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.

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]\displaystyle{ B_1 = {\alpha_0, \alpha_1, \ldots, \alpha_{m-1}} }[/math]

and

[math]\displaystyle{ B_2 = {\gamma_0, \gamma_1, \ldots, \gamma_{m-1}} }[/math]

then B₂ can be considered a dual basis of B₁ provided

[math]\displaystyle{ \operatorname{Tr}(\alpha_i\cdot \gamma_j) = \left\{\begin{matrix} 0, & \operatorname{if}\ i \neq j\\ 1, & \operatorname{otherwise} \end{matrix}\right. }[/math]

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

[math]\displaystyle{ \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

• Lidl, Rudolf; Niederreiter, Harald (1994). Introduction to finite fields and their applications. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139172769. ISBN 9781139172769. , Definition 2.30, p. 54.