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Thursday, January 21, 2016

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

Problem:


Solution:

Theorem 7 is the fundamental theorem of algebra, staying that any polynomial with coefficients in $ \mathbf{C} $ has a root in $ \mathbf{C} $.

Now suppose $ f $ has degree $ n $ with coefficients in $ \mathbf{C} $, but the fundamental theorem of algebra we have $ f $ have a root in $ \mathbf{C} $, so using the division theorem we know $ f = (x - a)g(x) $ with $ g(x) $ a polynomial of degree $ n - 1 $.

By induction we get the result we want.

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