Can you give me one example where the characteristic of a field is not equal to the number of elements in the field?
I know that the characteristic of a prime field $GF(p)$ is $p$, but even for a non-prime $q$, say $q = 6$, the characteristic is equal to 6.
So when the characteristic and the number of elements in a fields will be unequal?
Best Answer
The field $\Bbb{F}_{q}$ for $q=p^k$ is a field of characteristic $p$, but contains $p^k$ elements. To construct this you take $\Bbb{F}_p[x]/(m(x))$ for $m(x)$ a monic irreducible polynomial of degree $k$. Alternatively, this can be seen as the splitting field of $x^q-x\in \Bbb{F}_p[x]$.