There's this problem i have to solve, and I don't know if my approach is correct.(adsbygoogle = window.adsbygoogle || []).push({});

Problem:

Let K be an infinite field. Consider the ring R = K[X1,.....,Xn] of polynomials in n variables over K. Show that every polynomial in R gives rise to an unique function K^n --> K.

Idea to solution:

I know that for n=1, the lagrange interpolation formula gives this result. Then I'll try to use a proof with induction. Suppose you use the evaluation map on a variable Xi. The result gives you the ring in n-1 variables. By the induction hypotheses this ring gives rise to unique functions K^n-1 ---> K for every polynomial in this ring.

Now consider a polynomial P in R. Let's write the evaluation maps as f_i, this meaning the evaluation map in the variable X_i. Consider now the composition of g = f_1 o .... o f_n-1. By the induction hypothesis this a unique map from K^n-1 ---> K. We must now show that the map h = f_n o g, is unique. Essentially g is a polynomial in K[Xn], so by the lagrange interpolation formula this means that the evaluation map f_n is unique. So because g was unique, h = f_n o g is unique.

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I hope this approach is valid! Please give your comments on this!

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# Homework Help: Polynomial evaluation functions

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