Dual basis in a field extension

{{No inline|date=May 2025}}

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):

:$B\_1\; =\; \{\backslash alpha\_0,\; \backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_\{m-1\}\}$

and

:$B\_2\; =\; \{\backslash gamma\_0,\; \backslash gamma\_1,\; \backslash ldots,\; \backslash gamma\_\{m-1\}\}$

then B₂ can be considered a dual basis of B₁ provided

:$\backslash operatorname\{Tr\}(\backslash alpha\_i\backslash cdot\; \backslash gamma\_j)\; =\; \backslash begin\{cases\}$

0, & \operatorname{if}\ i \neq j

\\ 1, & \operatorname{otherwise}.

\end{cases}

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

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

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).