Irreducible over , yet reducible modulo every prime.
Counterexample bank
Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.
AllODEPDEabeliananalyticityanovabairecalculus of variationscanonical formscauchy-riemanncompactnesscompletenesscomplexconformalconnectednesscontinuitycontour integrationconvergencedifferentiationdistributionsfieldsfixed pointgaloisgreengroupsharmonicinjectivityintegral equationsintegrationlaurentlimsuplinear algebraliouvillelpmarkovmeasuremechanicsmetric spacesmultivariablemultivariatenumericalp-groupspermutationspolynomialspower seriesprobabilityquadratic formsreal analysisregressionresiduesringssamplingseparabilityseparationsequencesseriessingularitiesstabilitystatisticssylowtestingtopologyuniform convergencewronskianzeros
#1 · Complex Analysis, Algebra & Topology › Rings and Fields
“Irreducible over ℤ ⇒ irreducible mod every prime” — false
Counterexample locked — unlock with Notes + PYQ
polynomialsfields
#2 · Complex Analysis, Algebra & Topology › Rings and Fields
“A polynomial irreducible over ℚ is irreducible modulo some prime” — false
Counterexample: x⁴ + 1
polynomials
#3 · Complex Analysis, Algebra & Topology › Rings and Fields
“A degree-4 polynomial with no rational roots is irreducible over ℚ” — false
Counterexample locked — unlock with Notes + PYQ
polynomials