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

“Irreducible over ℤ ⇒ irreducible mod every prime” — false

Counterexample locked — unlock with Notes + PYQ

polynomialsfields

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

“An algebraic extension is a finite extension” — false

Counterexample: The field of all algebraic numbers Q̄ over ℚ

Every element is algebraic, but the extension has infinite degree (it contains $Q (2^{1/ n})$ for every n).

fields

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

“Every extension of degree n has a Galois group of order n” — false

Counterexample: ℚ(∛2)/ℚ has degree 3 but only the identity automorphism

The extension is not normal — the other cube roots are not real. Order = degree only for Galois extensions.

fieldsgalois