Modular invariant theory

In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by .
Dickson invariant.
When "G" is the finite general linear group GL"n"(F"q") over the finite field F"q" of order a prime power "q" acting on the ring F"q"["X"1, ...,"X""n"] in the natural way, found a complete set of invariants as follows. Write ["e"1, ..., "e""n"] for the determinant of the matrix whose entries are "X", where "e"1, ..., "e""n" are non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is
Then under the action of an element "g" of GL"n"(F"q") these determinants are all multiplied by det("g"), so they are all invariants of SL"n"(F"q") and the ratios ["e"1, ...,"e""n"] / [0, 1, ..., "n" − 1] are invariants of GL"n"(F"q"), called Dickson invariants. Dickson proved that the full ring of invariants F"q"["X"1, ...,"X""n"]GL"n"(F"q") is a polynomial algebra over the "n" Dickson invariants [0, 1, ..., "i" − 1, "i" + 1, ..., "n"] / [0, 1, ..., "n" − 1] for "i" = 0, 1, ..., "n" − 1.
 gave a shorter proof of Dickson's theorem.
The matrices ["e"1, ..., "e""n"] are divisible by all non-zero linear forms in the variables "X""i" with coefficients in the finite field F"q". In particular the Moore determinant [0, 1, ..., "n" − 1] is a product of such linear forms, taken over 1 + "q" + "q"2 + ... + "q""n" – 1 representatives of ("n" – 1)-dimensional projective space over the field. This factorization is similar to the factorization of the Vandermonde determinant into linear factors.