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"):
and
then "B"2 can be considered a dual basis of "B"1 provided
Here the trace of a value in GF("p""m") can be calculated as follows:
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).