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## Not all ideals are prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…

## The quaternion group is not a subgroup of Symmetric group for any n less than 8

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.7 Solution: (1) $Q_8$ is a subgroup of $S_8$ via the left regular representation. (2) Now suppose…

## Z[x] and Q[x] are not isomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.2 Prove that the rings $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. Proof: In $\mathbb{Q}[x]$, $f(x)+f(x)=g(x)$ has a…

## 2Z and 3Z are not isomorphic as rings

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.1 Prove that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic. Solution: Suppose \$\varphi : 2\mathbb{Z} \rightarrow…