Polynomial/K/Interpolation/Fact

Interpolation theorem for polynomials

Let $K$ be a field, and let $n$ different elements $a_1,\dots ,a_n\in K$, and $n$ elements $b_1,\dots ,b_n\in K$ be given.

Then there exists a unique

polynomial $P\in K[X]$ of degree $\le n-1$, such that $P\left(a_i\right)=b_i$

holds for all $i$.

Proof, Write another proof