## Definition and basic properties of the radical of an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.30 Solution: First, we certainly have $I \subseteq \mathsf{rad}(I)$ since for all $a \in I$, $a^1 \in…

## The set of nilpotent elements in a commutative ring is an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.29 Solution: Let $x,y \in \mathfrak{N}(R)$. Then for some nonnegative natural numbers $n$ and $m$, we have…

## Basic properties of the characteristic of a ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.26 Solution: (1) We begin by showing that $\varphi(a+b) = \varphi(a) + \varphi(b)$ for nonnegative $b$ by…

## The Binomial Theorem holds in any commutative ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.25 Solution: We begin with some lemmas. Recall that ${n \choose k} = \frac{n!}{k!(n-k)!}$, where $n$ is…

## 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…