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…
Prove that a given quotient ring is not an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.17 Solution: (1) Evidently, $$\overline{p(x)} = \overline{-x^2-11x+3},$$ $$\overline{q(x)} = \overline{8x^2 - 13x + 5},$$ $$\overline{p(x)+q(x)} = \overline{7x^2…
Compute in a quotient of a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.16 Solution: (1) Evidently, $$\overline{g(x)} = \overline{-2x^3 + 25936x + 3}.$$ (2) Note that $$x^4-16 = (x+2)(x-2)(x^2+4).$$…
A fact about ideals and subrings which intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: By the Second Isomorphism Theorem for rings, we have $$(S+I)/I \cong S/(S \cap I) =…