We know that all finite fields are extensions of fields . And in particular, any finite extension of them will not be algebraically closed. So the algebraic closure of is necessarily infinite.
Every finite field has order for some n. The underlying multiplicative group is always cyclic (of order ). Thus, for all .
Therefore, if , the equation will not have a root in .
If then this trick doesn't work...however, if is your finite field, then the polynomial has no roots in (this works for all p).