Algebra III - rings and fields

Fundamental concepts, constructions, and theorems:

• Rings, subrings, homomorphisms of rings
• Examples of rings
• Integral rings, fields of fractions
• Ideals, quotient rings and their universal properties, the characteristic of a ring, operations on ideals, correspondence theorems, product of rings, the Chinese remainder theorem
• Prime and maximal ideals

Arithmetics in rings

• Euclidean rings
• Principal rings
• Associated, prime, and irreducible elements
• Factorial rings
• Noetherian rings
• Characterization of factoriality
• Gauss's Lemma and theorem
• Irreducibility criteria

Field Theory:

• Algebras over a field
• Field extensions, algebraic and transcendental elements, the degree of a field extension, algebraic extensions, construction of simple field extensions, splitting fields
• finite fields
• separable extensions, primitive element theorem
• Galois theory
• purely inseparable extensions, separable-inseparable decomposition
• algebraically closed fields, separable closure, inseparable closure