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## A finite unital ring with no zero divisors is a field

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…

## In a unital ring, if the quotient corresponding to an ideal is a field, then the ideal is maximal

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…

## A commutative unital ring is a field precisely when the zero ideal is maximal

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…

## Verify that the set of complex numbers is a subfield of C

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 $F$…