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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