Ring Theory
Ring theory studies addition and multiplication through ideals, modules, localization, dimension, and homological structure.
Ramification theory, the largest field with about a quarter of the subject's articles, follows primes through finite extensions of Dedekind domains and discrete valuation rings: a prime is unramified exactly when it does not divide the discriminant, the norm of the different, and at an unramified prime with finite residue field the Frobenius element generates the decomposition group and is trivial exactly when the prime splits completely. In Dedekind domains the ideal class group is trivial exactly when the domain is a principal ideal domain, equivalently a unique factorization domain. Formal power series include Hahn series, and the subject also has Hilbert's syzygy theorem, the Quillen–Suslin theorem and the weak Nullstellensatz over \(\mathbb{C}\).
- Dedekind domains and ideal theory
- Ramification theory
- Valuation rings and Henselian rings
- Local rings, dimension and completion
- Ideals, localization and ring constructions
- Étale algebras, norms and differentials
- Syzygies and free resolutions
- Formal power series
- Modules over commutative rings
- Invariant theory
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