Tag
Number Fields I
7 resources tagged “Number Fields I” across the knowledge library.
7 items
Guide6 Aug 2026Algebraic Numbers and Number FieldsAlgebraic numbers and integers, minimal polynomials, number fields as finite extensions of Q, real and complex embeddings, the signature, and the primitive element theorem.5 min readRead MoreGuide6 Aug 2026Class Groups, Units and the RegulatorThe ideal class group, Dirichlet's unit theorem, the regulator, Minkowski's bound, and the analytic class number formula used to verify computed values.4 min readRead MoreGuide6 Aug 2026Decomposition of Prime Numbers in Number FieldsRamification indices and residue degrees, the fundamental identity, Dedekind's theorem relating prime decomposition to polynomial factorisation modulo p, and computing valuations.5 min readRead MoreGuide6 Aug 2026Discriminants and Integral BasesThe field discriminant, integral bases, the index of an equation order, and why computing the maximal order reduces to factoring the polynomial discriminant.5 min readRead MoreGuide6 Aug 2026Orders and Ideals in Number FieldsOrders, the maximal order, fractional and integral ideals, unique factorisation of ideals in a Dedekind domain, ideal arithmetic by Hermite normal form, and the two-element representation.4 min readRead MoreGuide6 Aug 2026Representing Algebraic NumbersThe standard, matrix, conjugate-vector and minimal-polynomial representations of an algebraic number, their relative costs, and how to convert between them.4 min readRead MoreGuide6 Aug 2026Trace, Norm and the Characteristic PolynomialDefinitions of trace and norm as sums and products over embeddings, their computation via the regular representation, multiplicativity, and their use in detecting units and computing ideal norms.4 min readRead More