Researched
Abstract Algebra (Noether)
Emmy Noether's 1921 work on rings and van der Waerden's Moderne Algebra (1930-31) recast algebra as the study of structures: groups, rings, fields and modules.
Open in the interactive tree →Noether's Idealtheorie in Ringbereichen (1921) analysed chains of ideals and gave Noetherian rings their name; van der Waerden's two-volume textbook made structures the organising idea of algebra. The language underlies algebraic number theory and algebraic geometry, and rings and finite fields carry modern cryptography.
Prerequisites
Unlocks
- Public-Key Cryptography1976RSA works in rings of integers mod n; key exchange and elliptic curves in finite groups and fields
- Fermat's Last Theorem Proved1995Wiles's proof uses elliptic curves and Galois representations, built on structural algebra
- Post-Quantum Cryptography2024ML-KEM and ML-DSA are lattice schemes over polynomial rings and modules