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Simply asking how can someone show that the vector space $V$ of all polynomials on a field, say $K$ cannot be generated with any finite set of vectors?

I don't know where to tackle the problem. :(

Thank you.

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    @JonasMeyer: Ops! Yes the second one is correct. I will fix it.2012-12-20

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Suppose that it could be, for vectors (polynomials) $p_1,\ldots, p_n$. Let $m=\text{max}\{\text{deg}(p):p=a_1p_1+\ldots+a_np_n:a_1,\ldots,a_n\in K\}.$ Surely a polynomial of degree $m+1$ exists in $V$. This is a contradiction.

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    @YACP I was thinking we'd still have to establish that no polynomial can exceed that degree in $V$ for the contradiction. But it is a simple proof. Same diff.2012-12-20
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Another possibility is to show that the infinite family $\{X^n : n\geq 0\}$ is linearly independent.