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We know that polynomial rings over $\mathbb{Q}$ is a vector space over $\mathbb{Q}$. It has a well-known basis $1, x, x^2,\ldots$ but can we classify all bases?

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    For @Rahman: What do you mean by classifying all bases?2012-11-24
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    Note that every basis has $\aleph_0$ members, but that means that there are $2^{\aleph_0}$ different bases. To see this note that for every $A\subseteq\mathbb N$ the set $\{x^n\mid n\in A\}\cup\{x^n+1\mid n\notin A\}$ is also a basis, and every $A$ generates a different basis.2012-11-24
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    I wonder whether this vector space has another maximal independent set except for standard basis.2012-11-24
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    Asaf just showed you a way to construct uncountably many different bases. Furthermore, it is straightforward to extend **any** finite independent set into a basis....2012-11-24

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