In an integral domain, there are at most two square roots of 1
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.11 Prove that if $R$ is an integral domain and $x^2 = 1$ for some $x \in…
The group of complex p-power roots of unity is a proper quotient of itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.8 Let $p$ be a prime and let $G$ be the group of $p-$power roots of 1 in…
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…
Compute the kernel and fibers of a given group homomorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.13 Let $G$ be the additive group of real numbers and $H$ the multiplicative group of complex…
Represent the real numbers as complex numbers of modulus 1
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.12 Let $G$ be the additive group of real numbers and $H$ the multiplicative group of complex…
Surjective group endomorphisms need not be automorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.19 Let $G = \{ z \in \mathbb{C} \ |\ z^n = 1\ \mathrm{for\ some}\ n \in…