Explore BrainMass
Share

Explore BrainMass

    Certai isomorphic modules

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Need stepwise solution of the attached problem.

    © BrainMass Inc. brainmass.com October 10, 2019, 7:00 am ad1c9bdddf
    https://brainmass.com/math/algebra/certai-isomorphic-modules-564384

    Attachments

    Solution Preview

    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.

    $2.19