Mathematics / advanced
Algebraic number theory
Replace unique factorization of elements with ideal arithmetic, local fields, ramification, and global invariants.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Algebraic number theory
Replace unique factorization of elements with ideal arithmetic, local fields, ramification, and global invariants.
- 01
Number fields and rings of integers
Find the integral arithmetic inside a finite rational extension and distinguish a power order from the full ring of integers.
35 min - 02
Algebraic integers, traces, and norms
Use multiplication maps and embeddings to calculate field trace and norm, keeping the ambient extension explicit.
35 min - 03
Dedekind domains and ideal factorization
Recover unique factorization for nonzero ideals while distinguishing irreducible elements, prime elements, and nonprincipal prime ideals.
38 min - 04
Discriminants and integral bases
Calculate trace-form discriminants and separate maximal-order ramification from polynomial discriminant artifacts caused by an index.
38 min - 05
Splitting and ramification of primes
Track ramification indices and residue degrees, and apply polynomial factorization only away from power-order index primes.
39 min - 06
Ideal class groups and Minkowski bounds
Measure the obstruction to principal ideals and use an explicit norm bound to reduce a class-number calculation to finitely many ideals.
40 min - 07
Dirichlet's unit theorem
Separate roots of unity from the free unit rank and interpret the logarithmic embedding with the correct complex weights.
38 min - 08
Completions and p-adic fields
Complete the rational numbers using a prime-adic metric and distinguish valuation normalization from finite residue precision.
39 min - 09
Hensel's lemma and local extensions
Lift simple modular roots in a complete local ring and distinguish unramified extensions from Eisenstein ramification.
41 min - 10
Decomposition, inertia, and Frobenius elements
Relate Galois stabilizers to residue-field automorphisms and fix arithmetic versus geometric Frobenius conventions.
41 min