## Compute in a polynomial ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.1 Let $p(x) = 2x^3 - 3x^2 + 4x - 5$ and let $q(x) = 7x^3 +…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.1 Let $p(x) = 2x^3 - 3x^2 + 4x - 5$ and let $q(x) = 7x^3 +…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu =…

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…

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…

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…

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…

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…

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…

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…

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 $n$…

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