Number fields expand on rational numbers, introducing algebraic elements. The ring of integers in these fields, denoted O_K, consists of all algebraic integers within the field. This concept generalizes regular integers to more complex number systems.

O_K forms a crucial structure in algebraic number theory. It's a subring of the number field, finitely generated as a Z-module. An integral basis for O_K allows representation of all its elements, facilitating calculations and deeper theoretical insights.

Ring of Integers in Number Fields

Definition and Characteristics

• Ring of integers O_K in a number field K encompasses all algebraic integers within K
• Algebraic integers constitute complex numbers serving as roots of monic polynomials with integer coefficients
• O_K forms a subring of K and an integral domain
• O_K exhibits finite generation as a Z-module, with rank matching the degree of K over Q
• Integral closure of Z in K manifests as O_K, incorporating all K elements integral over Z
• Quadratic number fields Q(√d) showcase specific O_K forms contingent on d ≡ 1 (mod 4) status
  • For d ≡ 1 (mod 4): O_K = Z[(1+√d)/2]
  • For d ≢ 1 (mod 4): O_K = Z[√d]
• O_K plays a pivotal role in algebraic number theory, extending the integer concept to number fields

Examples and Applications

• Gaussian integers Z[i] represent the ring of integers for Q(i)
• Eisenstein integers Z[ω], where ω is a cube root of unity, form the ring of integers for Q(√-3)
• Ring of integers for Q(√2) takes the form Z[√2]
• O_K facilitates the study of ideal factorization and class groups in number fields
• Prime factorization in O_K elucidates the behavior of rational primes in number field extensions

Integral Basis for Number Fields

Fundamental Concepts

• Integral basis {ω1, ..., ωn} for number field K comprises O_K elements forming a Z-basis for O_K
• Existence of integral basis stems from O_K's status as a free Z-module of rank n (n = degree of K over Q)
• Quadratic fields Q(√d) exhibit distinct integral bases based on d's congruence modulo 4
  • d ≡ 1 (mod 4): Integral basis {1, (1+√d)/2}
  • d ≢ 1 (mod 4): Integral basis {1, √d}
• Cubic fields often require discriminant and index formulas for integral basis determination
• Higher degree fields may necessitate advanced techniques (Round 2 algorithm, p-adic methods) for integral basis identification
• Integral basis enables representation of all O_K elements as Z-linear combinations of basis elements
• Non-uniqueness of integral basis persists, with unimodular transformations over Z relating different bases

Examples and Applications

• Q(√5) integral basis: {1, (1+√5)/2} (golden ratio appears)
• Q(√-7) integral basis: {1, √-7}
• Integral basis for Q(∛2): {1, ∛2, (∛2)²}
• Representation of elements using integral basis: α = a + b√d in Q(√d) when d ≢ 1 (mod 4)
• Integral basis facilitates norm and trace calculations in number fields

Discriminant of a Number Field

Definition and Properties

• Number field K discriminant defined as determinant of trace matrix for integral basis
• Discriminant formula: det(Tr(ωiωj)) for integral basis {ω1, ..., ωn}, Tr denoting trace function from K to Q
• Discriminant remains invariant across integral basis choices
• Absolute discriminant value gauges ring of integers "size" and informs K ramification
• Quadratic fields Q(√d) discriminant determination:
  • d ≡ 2,3 (mod 4): Discriminant = 4d
  • d ≡ 1 (mod 4): Discriminant = d
• Number field discriminant always yields non-zero integer
• Discriminant sign correlates with K's complex embedding count
• Minkowski's bound links discriminant to existence of small-norm non-trivial O_K elements

Calculation Examples and Applications

• Q(√5) discriminant: 5
• Q(√-7) discriminant: -28
• Q(∛2) discriminant: -108
• Discriminant aids in determining integral basis for number fields
• Ramification of primes in number field extensions relates to discriminant factorization
• Class number estimation employs discriminant in its calculations

Properties of the Ring of Integers

Dedekind Domain Characteristics

• Dedekind domain definition: Noetherian, integrally closed integral domain with maximal non-zero prime ideals
• O_K Noetherian property proof utilizes finite generation as Z-module
• O_K integral closure in fraction field K stems from definition as Z's integral closure in K
• Non-zero prime ideal maximality in O_K proven via finite field nature of prime ideal quotients
• Ascending chain condition on ideals satisfaction demonstrates O_K's Noetherian property
• Unique factorization of non-zero O_K ideals as prime ideal products exemplifies key Dedekind domain trait
• O_K fractional ideals form multiplication group, characteristic of Dedekind domains

Additional Properties and Examples

• O_K lacks unique factorization for elements in general (counterexample: Z[√-5])
• Ideal class group measures deviation from unique factorization in O_K
• Principal ideal domains (PIDs) form a subset of Dedekind domains (Z[i] as an example)
• O_K satisfies the Chinese Remainder Theorem for pairwise coprime ideals
• Localization of O_K at prime ideals yields discrete valuation rings