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This was given us as a fact, but why is this true? The zeropolynomial is the polynomial where all the coefficients are equal to $0$ if $R(x)$ is a polynomial over $\mathbb{C}$ and every $x\in \mathbb{N}_0$ is a zeropoint then one can rewrite the polynomial $R(x)=x(x-1)(x-2)…(x-n)…$ but if we put an $x\in \mathbb{C}-\mathbb{N}_0$ in $R(x)$, how does one know that $R(x)=0$?

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marked as duplicate by Dietrich Burde, Community Dec 16 '18 at 12:54

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    $\begingroup$ A polynomial of n-th degree has only n zero points. $\endgroup$ – Aretino Dec 16 '18 at 12:40
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A polynomial $f(x)$ has by definition a finite degree $n$ which is given by the highest degree $n$ of the variable $x$ involved in the polynomial. If you multiply two polynomials, the degrees add (if the underlying coefficient ring has no zero divisors as in the case of a field). Only the zero polynomial has all elements (in the coefficient ring or an extension ring) as zeros. A polynomial as described cannot exist.

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By the Fundamental theorem of algebra, a polynomial $p$ of degree $n>0$ has exactly $n$ complex roots (counting multiplicity). Let $ r_1,\dots,r_n$ be its root, then we must have $p(z)\ne 0$ for all $z$ such that $$ |z| > \max\{|r_1|,\dots,|r_n|\} =: R $$ since all the roots lie in the set $\{z\in\Bbb C: |z|\le R\}$.

This means that the only polynomial with infinitely many zeroes is the consstant polynomial $p\equiv0$.

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