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    Certai isomorphic modules

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    Dear student,

    See the attached file.

    Let us denote the K-module of all polynomials in K[x] of degree less than n by K_n[x].
    Let us construct the mapping f: K_n[x] → K^n as follows: f(a_0+a_1 x+⋯+a_(n-1) x^(n-1))=(a_0,a_1,...,a_(n-1)), for any polynomial 〖(a〗_0+a_1 x+⋯+a_(n-1) x^(n-1)).

    It is obvious that f is injective. Indeed, if 〖 polynomials (a〗_0+a_1 x+⋯+a_(n-1) ...

    Solution Summary

    It is proved that the K-module of polynomials of degree less than n is isomorphic to the module K^n.