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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 $$x^4-16 = (x+2)(x-2)(x^2+4).$$…

Compute the powers of a given ideal in a polynomial ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.36 Solution: We begin with a lemma. Lemma: If $R$ is a commutative ring with 1 and…

Characterize the units and nilpotent elements of a polynomial ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.33 Solution: (1) Note first that if $a_0$ is a unit in $R$ and $a_i$ nilpotent in…