# Doubt in section 4.3 of Hoffman and Kunze's *Linear Algebra*

In section 4.3 (Lagrange Interpolation) of Hoffman and Kunze's Linear Algebra, 2nd edition, on page 126 after the proof of Theorem 3, the authors write

From the results of the next section we shall obtain an altogether different proof of this theorem.

However, I cannot find another proof of Theorem 3 in the next section. Perhaps it is hidden in the results proved there, but I am not able to deduce any such proof from those results. Can someone help me find this different proof of Theorem 3?

The statement of the theorem in question is as follows:

Theorem 3. If $F$ is a field containing an infinite number of distinct elements, the mapping $f \to f^\sim$ is an isomorphism of the algebra of polynomials over $F$ onto the algebra of polynomial functions over $F$.

The first proof is based on Lagrange's interpolation formula. But you can also prove the theorem by using this result on page 129:

Corollary 2: A polynomial $f$ of degree $n$ over a field $F$ has at most $n$ roots in $F$.

Proof. Let $f$ and $g$ be two polynomials such that $f^\sim = g^\sim$. Setting $h = f - g$, one gets $h^\sim = (f-g)^\sim = f^\sim - g^\sim = 0$. In particular $h^\sim(x) = 0$ for all $x \in F$ and since $F$ is infinite, $f$ has infinitely many roots, and hence is the null polynomial.

• After proving the above corollary, the authors state "The reader should observe that the main step in the proof of Theorem 3 follows immediately from this corollary." I somehow managed to miss this line completely. But thank you for the details!
– user279515
Commented Jan 11, 2018 at 10:06
• You're welcome. Commented Jan 11, 2018 at 11:01