I have this problem that i need to prove and i dont even know where to start. So I have to show that the symmetric algebra ( Sym V ) is isomorphic to free commutative R-algebra on the set {x1, ..., xn}.(adsbygoogle = window.adsbygoogle || []).push({});

Now i know that Sym V could be regarded as +Sym^n V for n>=0. And then we need to show that this is isomorphic to the R-algebra on the set {x1, ..., xn}. What I dont really understand is the +Sym^n V for n>=0

what exactely does the + do? Can i just regard this as the space of homogeneous polynomials of degree n in the variables of e1, e2,... e3, where {ei} is a basis for V.

Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. Do I need to find a map such that it reduces all the polynomials from R to make them homogeneous.. I really dont know how to start this problem.

thnx for everyone who can help me with any ideas:)

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# Symmetric powers and R-algebra

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