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):

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

and

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

then B₂ can be considered a dual basis of B₁ provided

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

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

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

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).

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.