Dual basis in a field extension

Template:No inline 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 is true if L is separable over K; it is always true if K is a perfect field, including when K is finite or of characteristic zero.

A dual basis is not a specific basis like the polynomial basis or the normal basis; rather for any given basis, it is another associated basis which is useful for computations.

We say that two bases of a finite field $L=\text{GF}(p^n)$ over $K=\text{GF}(p)=\mathbb{Z}/p\mathbb{Z}$,

$B_1=\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{n-1}\},B_2=\{{\beta }_0,{\beta }_1,\dots ,{\beta }_{n-1}\},$

are dual to each other provided

$\mathrm{Tr}({\alpha }_i\cdot {\beta }_j)=\{\begin{array}{cc}1& \text{if }i=j\\ 0& \text{if }i\ne j.\end{array}$

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

$\mathrm{Tr}(\gamma )=\sum _{i=0}^{n-1}{\gamma }^{p^i}$

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

• Template:Cite book, Definition 2.30, p. 54.