What is the characteristic of field $GL(n,q)$ where $q$ is prime?
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5$GL(n,q)$ is not a field but a group; the notion of characteristic does not apply. – Marc van Leeuwen Jul 29 '13 at 07:09
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Maybe you meant the finite field $GF(q^n)$ instead of the group $GL(n,q)$; for that case at least the question makes sense. Since the field $GF(q^n)$ contains $GF(q)$ as a subfield, it has the same characteristic as that subfield. Given that $q$ is prime, that characteristic is$~q$.
Marc van Leeuwen
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The notation $M_n(q^n)$ is not standard as far as I know. In the parentheses you need a ring rather than a number. And the characteristic of $M_n(\cdot)$ will be the same as the ring you put at the dot. If the ring is the field $GF(q^n)$ (also written $\Bbb F_{q^n}$), then its characteristic is $q$ (assuming $q$ was a prime), but if it is the ring $\Bbb Z/q^n\Bbb Z$ (also written by some as $\Bbb Z_{q^n}$) then its characteristic is $q^n$, – Marc van Leeuwen Jul 29 '13 at 17:32