Moore determinant

A determinant, named after E. H. Moore, defined over a finite field from a square Moore matrix, analogous to the Vandermonde determinant. A Moore matrix has successive powers of the Frobenius automorphism applied to the first column, i.e., an $m \times n$ $$ M=\begin{bmatrix} \alpha_1 & \alpha_1^q & \dots & \alpha_1^{q^{n-1}}\\ \alpha_2 & \alpha_2^q & \dots & \alpha_2^{q^{n-1}}\\ \alpha_3 & \alpha_3^q & \dots & \alpha_3^{q^{n-1}}\\ \vdots & \vdots & \ddots &\vdots \\ \alpha_m & \alpha_m^q & \dots & \alpha_m^{q^{n-1}}\\ \end{bmatrix} $$ or $$ M_{i,j} = \alpha_i^{q^{j-1}} $$ for all indices $i$ and $j$. (Some authors use the transpose of the above matrix.)

The set $\{\alpha_1,\ldots,\alpha_m\}$ is a basis for $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ if and only if the corresponding Moore determinant is non-zero.

The Moore determinant can be expressed as: $$ \det(V) = \prod_{\mathbf{c}} \left( c_1\alpha_1 + \cdots c_n\alpha_n \right) \,, $$ where $c$ runs over a complete set of direction vectors, made specific by having the last non-zero entry equal to 1.

References

• David Goss, "Basic Structures of Function Field Arithmetic", Springer Verlag (1996) ISBN 3-540-63541-6. Chapter 1.
• Gary L. Mullen, Daniel Panario (edd), "Handbook of Finite Fields", CRC Press (2013) ISBN 1439873828