Is Lagrange interpolation formula not unique over a finite field?

Question

Linear Algebra - Friedberg p.53

Let $F$ be a field and $c_0,\ldots,c_n$ be distinct scalars in $F$. Then, define $l_j(X)=\prod_{i=0\bigwedge i\neq j}^n \frac{X-c_i}{c_j-c_i}$ (Lagrange Polynomials). Then $\{l_0(X),\ldots,l_n(X)\}$ is a basis for $P_n(F)$.

If $F$ is infinite, then it has following properties;

1. For each $f(X)\in P_n(F), f(X)=\sum_{i=0}^n f(c_i)l_i(X)$.

2. If $f(X),g(X)\in P_n(F)$ and [$\forall 0≦i≦n,\ f(c_i)=g(c_i)$], then $f(X)=g(X)$.

I don't know why infiniteness of $F$ should be in the hypothesis. I cannot find where i used the hypothesis, that is, " $\forall a\in F,\ f(a)=g(a) \Rightarrow f(X)=g(X)$ ".

Answer 1

I suppose $P_{n}(F)$ denotes the set of polynomials in $F[X]$ of degree at most $n$.

You are right, the statements hold for any field $F$, finite or infinite.

In fact, the first statement follows from the second one, and the second holds because in a field a nonzero polynomial of degree $n$ can have at most $n$ distinct roots.