Tags: #qual_algebra

Week 4: Rings

• Morphisms, Ideals, quotients, zero divisors, isomorphism theorems, CRT
• Irreducible and prime elements, nilpotent, units
• Radical, nilradical, spec and maxspec
• Special types: domains, integral domains, Euclidean ⇒ PID ⇒ UFD ⇒?, Dedekind domains, Noetherian, Artinian
• Zorn’s lemma arguments
• Bonus optional stuff: localization

Prove that a commutative ring with unit is a field if and only if its only ideals are {0} and the whole ring

Show the irreducibility criterion for polynomials \(f\in k[x]\) of degree 2 or 3: such a polynomial is irreducible iff it has no roots in the field k