Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.4 Prove that if $R$ is an integral domain then the ring $R[[x]]$ of formal power series…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.3 Let $R$ be a ring. Define the set $R[[x]]$ of formal power series in the indeterminate…
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 +…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.30 Let $A = \prod_\mathbb{N} \mathbb{Z}$ be the direct product of countably many copies of $\mathbb{Z}$. Recall…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.29 Let $A$ be any abelian group. Let $R = \mathsf{Hom}(A,A)$ be the set of all group…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.28 Let $R$ be a ring with $1 \neq 0$. A nonzero element $a \in R$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.27 A specific example of a discrete valuation ring is obtained when $p$ is a prime, $K…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.26 Let $K$ be a field. A discrete valuation on $K$ is a function $v : K^\times…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.25 Let $I$ be the ring of integral Hamiltonian Quaternions and define $N : I \rightarrow \mathbb{Z}$…