The ring homomorphic image of an ideal is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.24 Solution: (1) Let $x,y \in \varphi^\ast[J]$. Now $0 \in J$ and $\varphi(0) = 0$, so that…
Exhibit a group homomorphism from Z/(8) to Z/(4)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.10 Let $\varphi : \mathbb{Z}/(8) \rightarrow \mathbb{Z}/(4)$ be defined by $\overline{a} \mapsto \overline{a}$. (Note that $\overline{a}$ means…
Describe the kernel and fibers of a given group homomorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.9 Define $\varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ by $$a+bi \mapsto a^2 + b^2.$$ Prove that $\varphi$ is…
Absolute value is a group homomorphism on the multiplicative real numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.8 Let $\varphi : \mathbb{R}^\times \rightarrow \mathbb{R}^\times$ be given by $x \mapsto |x|$. Prove that $\varphi$ is…
The image of a group homomorphism is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.13 Let $G$ and $H$ be groups and let $\varphi : G \rightarrow H$ be a group…