Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.11 Solution: We know from calculus that \begin{align*}\varphi(f+g) =&\ \int_0^1 (f+g)(x) dx\\ =&\ \int_0^1 f(x) + g(x)…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.14 Consider the additive quotient group $\mathbb{Q}/\mathbb{Z}$. (1) Show that every coset of $\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$ has…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.11 Let $F$ be a field and let $$G = \left\{ \begin{bmatrix} a & b \\ 0…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.10 Let $\varphi : \mathbb{Z}/(8) \rightarrow \mathbb{Z}/(4)$ be defined by $\overline{a} \mapsto \overline{a}$. (Note that $\overline{a}$ means…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.9 Define $\varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ by $$a+bi \mapsto a^2 + b^2.$$ Prove that $\varphi$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.7 Define $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $\pi(x,y) = x + y$. Prove that $\pi$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.6 Define $\varphi : \mathbb{R}^\times \rightarrow \{ \pm 1 \}$ by $x \mapsto x / |x|$. Describe…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.2 Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.1 Let $\varphi : G \rightarrow H$ be a group homomorphism and let $E \leq H$ be…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.19 Show that if $H$ is a group and $h \in H$, there exists a unique homomorphism…