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Compute in a quotient of a polynomial ring


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.16

Solution:

(1) Evidently, $$\overline{g(x)} = \overline{-2x^3 + 25936x + 3}.$$ (2) Note that $$x^4-16 = (x+2)(x-2)(x^2+4).$$ By Exercise 7.4.14, $\overline{x+2}$ and $\overline{x-2}$ are zero divisors in $\overline{E}$.


Linearity

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