Polynomial/K/Interpolation/Fact/Proof

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Proof

We prove the existence and consider first the situation where for all for some fixed . Then

is a polynomial of degree which at the points has value . The polynomial

has at these points still a zero, but additionally at its value is . We denote this polynomial . Then

is the polynomial looked for, because for the point we have

for and .

The uniqueness follows from fact.