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## Every subring of a field which contains 1 is an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.12 Prove that any subring of a field which contains the identity is an integral domain. Solution:…

## The center of a ring is a subring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.7 The center of a ring $R$ is Z(R) = \{ z \in R \ |\ zr…

## General linear groups of dimension at least 2 are nonabelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.8 Show that $GL_n(F)$ is nonabelian for all $n \geq 2$ and all fields $F$. Solution: Recall…

## Compute the center of a Heisenberg group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.14 Let $F$ be a field and let $H(F)$ denote the Heisenberg group over $F$ as defined…