# Basis of Polynomials over a Finite Field

I'm trying to show that $1, x, x^2, .... x^k$ are a basis for the space of all polynomials with degree less than or k over a finite field $\mathbb{K}$ with p elements. I know that over the reals I would take $a_1 + a_2x + ... a_kx^k = 0$, let $x = 0$ to show $a_1=0$, take the derivative, and continue on in a similar way, but I don't see how to show independence over a finite field.

Over any field $F$, these polynomials form a basis for the mentioned vector space. They clearly span the vector space. To show they are independent you need to realize that a polynomial in the context of algebra is not a function. Two polynomials over a field $F$ are equal if, and only if, their coefficients are identical. This immediately establishes that the set of polynomials mentioned is an independent set.
The confusion arises since when the field is $\mathbb R$ (or more generally, for any infinite field) a polynomial induces a unique polynomial function. That means that functional methods can be used to proof independence (this is essentially the argument you supplied). But, when the field $F$ is finite, two different polynomials may give rise to the exact same polynomial function. In analysis we would consider these two polynomials as the same, but in algebra we don't. For example, the polynomials $X^n + X$, for $n\ge 1$, when considered as polynomials with coefficients in $\mathbb Z_2$, are all different, yet they all induce the same polynomial function (check which one).