# Algebra Qual Prep Week 4

Tags: #qual_algebra

# Week 4: Rings

• Morphisms, Ideals, quotients, zero divisors, isomorphism theorems, CRT
• Irreducible and prime elements, nilpotent, units
• Special types: domains, integral domains, Euclidean ⇒ PID ⇒ UFD ⇒?, Dedekind domains, Noetherian, Artinian
• Zorn’s lemma arguments
• Bonus optional stuff: localization
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Prove that a commutative ring with unit is a field if and only if its only ideals are {0} and the whole ring

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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

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