Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.13 Solution: Define $$\varphi : \mathbb{C} \rightarrow M_2(\mathbb{R}) by a+bi \mapsto \begin{bmatrix} a & b \\ -b…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.7 Let $F$ be a field and $V$ an $F$-vector space. The axioms for vector spaces over…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.1 Solution: Let $a \in F$. We have $1 \cdot a = 1a = a$ by the…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.12 Prove that any subring of a field which contains the identity is an integral domain. Solution:…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.7 The center of a ring $R$ is $$Z(R) = \{ z \in R \ |\ zr…
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…
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…