Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.21 Solution: [We will assume Wedderburn’s Theorem.] Let $R$ be a finite ring (not necessarily commutative) with…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.20 Solution: Let $R$ be a finite commutative ring with no zero divisors. Now let $I \subseteq…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.18 Solution: We first prove a lemma. Lemma: The map $\varphi : R[[x]] \rightarrow R$ given by…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.8 Solution: Call the additive and multiplicative identities of $\mb F$ $0_{\mb F}$ and $1_{\mb F}$ respectively. Define $n_{\mb F}$…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.7 Solution: Every subfield of $\C$ has characterisitc zero since if $\mb F$ is a subfield then $1\in \mb F$…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.5 Solution: We must check the nine conditions on pages 1-2: An operation is commutative if the table is symmetric…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.7 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Then $\varphi : R[x]…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.5 Solution: Suppose we have a two-sided ideal $I$ with $M \subseteq I \subseteq R$. By the…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.4 Solution: ($\Rightarrow$) Suppose $R$ is a field. Let $I \subseteq R$ be an ideal which properly…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.1 Solution: Let $F=\{x+y\sqrt{2}\mid x,y\in\mb Q\}$. Then we must show six things: $0$ is in $F$ $1$ is in $F$…