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## Verify that the set of complex numbers is a subfield of C

Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.1 Solution: Let $F=\{x+y\sqrt{2}\mid x,y\in\mb Q\}$. Then we must show six things: $0$ is in $F$ $1$ is in $F$…

## Embed the complex numbers in a ring of real matrices

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…

## 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…