If $\mathbb{F}=\{ 0,1,x,y\}$ is a field with four elements, why is the characteristic 2?
In a field with four elements, the characteristic is 2 or 3(since it has to be a prime). Is there any way to determine precisely the characteristic of this field, and in general any finite field?
Best Answer
If a field $\mathbb{F}$ contains a field $\mathbb{K}$, then $\mathbb{F}$ is a vector space over $\mathbb{K}$.
A field of $4$ elements cannot contain a field of $3$ elements, because then it would be a vector space over $\mathbb{F}_3$, and so would necessarily have cardinality $3^k$ for some $k$. But $4\neq 3^k$.
(Alternatively: the additive subgroup generated by $1$ must have order dividing the order of the field, by Lagrange's Theorem. So the subfield generated by $1$ must have order $2$; it cannot have order $3$.)
A finite field must be a vector space over the field generated by $1$; hence its order will be $p^k$ for some prime $p$ and some positive integer $k$, and the characteristic will then be $p$.