Compute in a polynomial ring
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 +…
Counterexamples regarding one-sided zero divisors and inverses
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…
The set of all endomorphisms of an abelian group is a ring under pointwise addition and composition
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…
Basic properties of left and right units and left and right zero divisors
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…
Prove that a given function is a valuation and find its corresponding valuation ring
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…
Basic properties of discrete valuations
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…
A norm on the integral Hamiltonians
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 $N : I \rightarrow \mathbb{Z}$…
Construct elements of infinite multiplicative order in some quadratic integer rings
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.24 Show that for $D \in \{ 3,5,6,7 \}$ the group of units in $\mathbb{Z}[\omega]$ is infinite…
The order of conductor f in the ring of quadratic integers is a subring and has index f
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.23 Let $D$ be a squarefree integer, and let $O$ be the ring of integers in the…
Infinite Boolean rings exist
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.22 Give an example of an infinite boolean ring. Solution: If $X$ is an infinite set, then…