Unit · year 3

MU-303 · Rings, Fields & Galois Theory

Threads structure26 lectures6 theorems

From factorisation in rings to the symmetry theory that settles the quintic.

PREREQUISITES

L01	Rings: Axioms and Examples —
L02	Ideals and Quotient Rings —
L03	Ring Homomorphisms and the Isomorphism Theorems —
L04	Integral Domains and Fields —
L05	Polynomial Rings —
L06	Divisibility, Primes, and Irreducibles —
L07	Euclidean Domains Every PID is a UFD
L08	Principal Ideal Domains Every PID is a UFD
L09	Unique Factorisation Domains Every PID is a UFD
L10	Every PID Is a UFD Every PID is a UFD
L11	Gauss's Lemma and ℤ[x] Every PID is a UFD
L12	Tests for Irreducibility Eisenstein's criterion
L13	Eisenstein's Criterion Eisenstein's criterion
L14	Field Extensions and Degree The tower law
L15	The Tower Law The tower law
L16	Algebraic and Transcendental Elements The tower law
L17	Splitting Fields —
L18	Ruler-and-Compass Constructions The tower law
L19	Finite Fields Classification of finite fields
L20	Classification of Finite Fields Classification of finite fields
L21	Separability and Normality —
L22	The Galois Group The fundamental theorem of Galois theory
L23	The Galois Correspondence The fundamental theorem of Galois theory
L24	Solvability by Radicals The fundamental theorem of Galois theory, Insolvability of the quintic
L25	Insolvability of the Quintic Insolvability of the quintic
L26	Synthesis: Symmetry Decides Solvability The fundamental theorem of Galois theory, Insolvability of the quintic