Polynomial/K/Interpolation/Fact

Interpolation theorem for polynomials

Let ${}K$ be a field, and let ${}n$ different elements ${}a_{1},\ldots ,a_{n}\in K$ and ${}n$ elements ${}b_{1},\ldots ,b_{n}\in K$ are given.

Then there exist a unique

polynomial ${}P\in K[X]$ of degree ${}\leq n-1$, such that ${}P{\left(a_{i}\right)}=b_{i}$

holds for all ${}i$.

Proof, Write another proof