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The group of units in $\mathbb Z/(2^n)$ is not cyclic for n at least 3

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.23 Show that $(\mathbb{Z}/(2^n))^\times$ is not cyclic for any $n \geq 3$. (Hint: find two distinct subgroups…

Compute the order of 5 in the integers mod a power of 2

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.22 Let $n$ be an integer with $n \geq 3$. Use the Binomial Theorem to show that…

Use the Binomial Theorem to compute the order of an element in the integers mod a prime power

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.21 Let $p$ be an odd prime and $n$ a positive integer. Use the Binomial Theorem to…

Compute the order of a cyclic subgroup in Z/(54)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.10 What is the order of $\overline{30}$ in $\mathbb{Z}/(54)$? Write out all of the elements and their…

Find all generators of Z/(202)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.4 Find all generators for $\mathbb{Z}/(202)$. Solution: The generators of $\mathbb{Z}/(202)$ are precisely those (equivalence classes represented…

Find the generators of Z/(48)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.3 Find all the generators in $\mathbb{Z}/(48)$. Solution: The generators of $\mathbb{Z}/(48)$ are precisely those (equivalence classes…

Compute the subgroup lattice of Z/(45)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.1 Find all subgroups of $G = \mathbb{Z}/(45)$, giving a generator for each. Describe the containments among…

If n is composite, then Z/(n) is not a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4 Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field. Solution: If $n$…