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Saturday, February 6, 2016

UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 9

Problem:


Solution:

Because the polynomial with a single variable is a principal ideal domain. The ideal $ \langle x^3 + x^2 - 4x - 4, x^3 - x^2 - 4x + 4, x^3 - 2x^2 - x + 2 \rangle $ is essentially generated by their GCD, so we find the GCD of these polynomials using the program we developed for the last problem, it turns out to be $ \langle x - 2 \rangle $.

Now it is obvious that $ x^2 - 4 = (x + 2)(x - 2) \in \langle x - 2 \rangle = \langle x^3 + x^2 - 4x - 4, x^3 - x^2 - 4x + 4, x^3 - 2x^2 - x + 2 \rangle $.

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