Consider V a vector space and the symmetric algebra $S(V^*)$ is it possible to define the polynomial on $V$, $R[V]$ canonically ? I.e. without a use of base ? And show this is isomorphic to the symmetric algebra ?
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$\begingroup$ Well, what prevents you from defining polynomial functions on $V$ as $S(V^*)$? It clearly makes sense: $V^*$ consists of all linear functions, and multiplying them to to create polynomials is a natural thing. Or are you worried about the possible side effects in positive characteristic? $\endgroup$– Vladimir DotsenkoCommented Jun 8, 2011 at 21:23
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1$\begingroup$ Polynomials on $V$ are precisely elements of $S(V^{\ast})$ by the universal property of the symmetric algebra. There are no characteristic issues here. $\endgroup$– Qiaochu YuanCommented Jun 8, 2011 at 21:33
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This has come up several times before:
Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces
Symmetric powers and duals of vector bundles in char p
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
answered Jun 8, 2011 at 21:28