Then under the action of an element ''g'' of GL<sub>''n''</sub>('''F'''<sub>''q''</sub>) these determinants are all multiplied by det(''g''), so they are all invariants of SL<sub>''n''</sub>('''F'''<sub>''pq''</sub>) and the ratios [''e''<sub>1</sub>, ...,''e''<sub>''n''</sub>]/[0, 1, ...,''n'' − 1] are invariants of GL<sub>''n''</sub>('''F'''<sub>''q''</sub>), called '''Dickson invariants'''. Dickson proved that the full ring of invariants '''F'''<sub>''q''</sub>[''X''<sub>1</sub>, ...,''X''<sub>''n''</sub>]<sup>GL<sub>''n''</sub>('''F'''<sub>''q''</sub>)</sup> 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.
{{harvtxt|Steinberg|1987}} gave a shorter proof of Dickson's theorem.
