Mathematics / core
Rings, ideals, and polynomial arithmetic
Extend arithmetic to rings and polynomial systems while tracking ideals, factorization, and quotient constructions.
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Inside this module
Module 01 core
Rings, ideals, and polynomial arithmetic
Extend arithmetic to rings and polynomial systems while tracking ideals, factorization, and quotient constructions.
- 01
Rings, units, and zero divisors
Determine which familiar arithmetic operations survive in a ring, and distinguish invertibility from cancellation.
24 min - 02
Ideals and quotient rings
Construct arithmetic on equivalence classes and see why multiplication requires an ideal, not merely an additive subgroup.
26 min - 03
Ring homomorphisms and isomorphisms
Use kernels and images to turn a structure-preserving map into an exact description of a quotient ring.
25 min - 04
Integral domains and fraction fields
Construct fractions from a domain and identify the exact role of nonzero denominators and cancellation.
27 min - 05
Polynomial division and root multiplicity
Derive remainders, root bounds, and repeated roots from division by a polynomial with invertible leading coefficient.
28 min - 06
Euclidean domains and principal ideals
Explain why a remainder algorithm produces greatest common divisors, Bézout coefficients, and a generator for every ideal.
29 min - 07
Unique factorization domains
Separate irreducible factors from prime divisors, and understand what uniqueness means when units and associates are present.
29 min - 08
Irreducibility and Gauss's lemma
Move polynomial factorization between integers, rationals, and finite fields without losing coefficient assumptions.
30 min - 09
Chinese remainders for rings
Split a quotient into compatible local components, prove the comaximal condition, and reconstruct an element explicitly.
29 min - 10
Polynomial interpolation over fields
Construct a polynomial from prescribed values, prove uniqueness, and connect interpolation to a quotient-ring decomposition.
30 min