21-373 Algebraic Structures

9 units, Carnegie Mellon University

Groups: Homomorphisms. Subgroups, cosets, Lagrange's theorem. Conjugation. Normal subgroups, quotient groups, first isomorphism theorem. Group actions, Cauchy's Theorem. Dihedral and alternating groups. The second and third isomorphism theorems. Rings: Subrings, ideals, quotient rings, first isomorphism theorem. Polynomial rings. Prime and maximal ideals, prime and irreducible elements. PIDs and UFDs. Noetherian domains. Gauss' lemma. Eisenstein criterion. Fields: Field of fractions of an integral domain. Finite fields. Applications to coding theory, cryptography, number theory. (Three 50 minute lectures)

Prerequisites: (21-127 or 21-128 or 15-151) and (21-242 or 21-241)

Offered: Spring 2027, Fall 2026, Summer 2026, Spring 2026, Fall 2025

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