irreducibility criteria over function fields

Question

Can any one help me what it means a polynomial is irreducible over function field over finite field with one example. Thank you very much.

Answer 1

Let $\mathbb F_5$ be the field of order $5$ and let $K=\mathbb F_5(X)$ be the function field over $\mathbb F_5$. Consider the polynomial $$P(T)=T^2 - \dfrac{1}{X}\in K[T]$$ with coefficients in $K$.

$P$ is irreducible over $K$ if and only if $P$ has no roots in $K$ (since $P$ is degree $2$).

Let us see by contradiction that $P$ is irreducible in $K$. Assume that $P$ has a root $\alpha$ in $K$. Then there are polynomials $Q(X), R(X)$ with coefficients in $\mathbb F_5$ so that $\alpha = \dfrac{Q(X)}{R(X)}$ and $\alpha^2 = \dfrac{1}{X}$. Equivalently $$ X\cdot Q(X) ^2 = R(X)^2 $$ as an equality in $\mathbb F_5[X]$. Left hand side has odd degree and right hand side has even degree.

Edit: Maybe it's important to remark that this example makes no use of properties of $\mathbb F_5$ other than being a field. So formally speaking this doesn't show any of the cool things of finites fields or positive characteristic fields.