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## Count the number of cyclic subgroups in an elementary abelian p-group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.11 Solution: By definition, every nonidentity element of $E_{p^n}$ has order $p$. Thus every nonidentity element generates…

## The characteristic of an integral domain is prime or zero

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.28 Solution: Suppose the characteristic $n$ of $R$ is composite, and that $n = ab$ where $a$…

## n divides the totient of $p^n-1$ when p is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Let $p$ be a prime and let $n$ be a positive integer. Find the order of…

## Alternate proof of Cauchy’s Theorem

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.9 Solution: (1) We prove this equality by attempting to choose an arbitrary element of $\mathcal{S}$. Note…

## Examples of nilpotent elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.13 An element $x \in R$, $R$ a ring, is called nilpotent if $x^m = 0$ for some…