Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every integral domain is a UFD” — false

Counterexample: ℤ[√−5]: 6 = 2·3 = (1+√−5)(1−√−5)

rings

#2 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every UFD is a PID” — false

Counterexample locked — unlock with Notes + PYQ

rings

#3 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every PID is Euclidean” — false

Counterexample locked — unlock with Notes + PYQ

rings

#4 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every prime ideal is maximal” — false

Counterexample: (X) in ℤ[X], or (0) in any integral domain that is not a field

$Z [X] / (X) ≅ Z$ is a domain but not a field. In a PID the implication does hold for non-zero primes.

rings

#5 · Complex Analysis, Algebra & Topology › Rings and Fields

“In every integral domain, irreducible implies prime” — false

Counterexample: 2 in ℤ[√−5]

2 is irreducible (no element of norm 2) but divides $(1 + \sqrt - 5) (1 - \sqrt - 5) = 6$ without dividing either factor.

rings

#6 · Complex Analysis, Algebra & Topology › Rings and Fields

“R a PID implies R[X] is a PID” — false

Counterexample locked — unlock with Notes + PYQ

rings