Finite fields as splitting fields

Question

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the polynomial $t^{5^2} - t$. Can somebody explain how they make this jump. Thanks in advance.

Answer 1

The zeros of $t^{25} - t$ in some algebraic closure of $\mathbb{F}_5$ form a field (due to the "Freshman's dream" $$(a+b)^{25} = a^{25} + b^{25}$$ in fields of characteristic $5$), and up to isomorphism, all finite fields of the same cardinality are the same. Both fields in question have $25$ elements - they must be the same.

You can make this explicit, though. Letting $\alpha$ be a fixed zero of $t^2 - 2$, all of the zeros of $t^{25} - t$ lie in $\mathbb{F}_5(\alpha)$: they are the elements $c\alpha + d$ with $c,d \in \mathbb{F}_5$. There are $25$ such elements and, again by the Freshman's dream, they are all actually zeros, so that's all of them.