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Our professor gave us as an exercise (for the first part) to find a ring structure on the set of a finite sequence of elements in a field $K$ such that the set of these sequences be isomorphic to $K[x]$. This was easy. But the second part of the exercise said: "Expand this statement to the set of all sequences of elements in $K$".

Now my question is: Was he maybe messing with us? Because, as far as I got about thinking about this exercise, even $K[x_1]\cdots [x_n]$ can be bijectively mapped only to finite sequences, so considerung (infinite) sequences would get me nowhere.

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$K[x_1]\cdots[x_n] \cong K[x_1, \ldots, x_n]$, and is not the answer your professor is looking for. Why not think about $K[x]$ a little more — what happens if we allow polynomials with infinitely many non-zero coefficients? –  Zhen Lin Nov 6 '11 at 17:25

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Have you heard of the ring of power series, which in your case would be $k[[x]]$? As the name suggests, this is similar, yet different from power series you study in real and complex analysis. It is the algebraic analogue of power series studied in real and complex analysis, but with coefficients in an arbitrary ring. I am presuming since your professor assigned this question, he has not taught power series in class yet. So, it might be a good idea to figure the question out for yourself first, and then look up power series rings. So, long story short, your professor is not messing with you.

Also, I like Lang's section on power series. So, if you need a reference, you could take a look there (book is Algebra by Serge Lang).

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Consider the ring of formal series $K[[x]]$.

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