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 $$x^4-16 = (x+2)(x-2)(x^2+4).$$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.14 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring with $1 \neq…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.7 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Then $\varphi : R[x]…
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…
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…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.10 Solution: (1) We claim that this subset $S$ is an ideal. To that end, suppose $\alpha…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.2 Prove that the rings $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. Proof: In $\mathbb{Q}[x]$, $f(x)+f(x)=g(x)$ has a…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.6 As in Exercise 2.2.12, let $S_4$ act on the set $R$ of all polynomials with integer…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.2 Let $R$ be a commutative ring with $1 \neq 0$. Let $p(x) = \sum_{i=0}^n a_ix^i$ be…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.1 Let $p(x) = 2x^3 - 3x^2 + 4x - 5$ and let $q(x) = 7x^3 +…