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## Exhibit a mapping which is a group homomorphism but not a ring homomorphism

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)…

## Q/Z is isomorphic to the group of complex roots of unity

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…

## Describe the fibers of a given group homomorphism

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…

## Sign map is a homomorphism on the multiplicative group of reals

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…