6.3 Ideal arithmetic and operations

Ideal arithmetic in Dedekind domains is all about adding, multiplying, and dividing ideals. These operations let us work with more complex algebraic structures and solve tricky number theory problems.

Understanding ideal arithmetic helps us factor ideals into primes, study how primes behave in number fields, and calculate important values like class numbers. It's a key tool for tackling advanced topics in algebraic number theory.

Arithmetic of Ideals in Dedekind Domains

Definition of Ideal Operations

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• Dedekind domains represent integral domains where every non-zero proper ideal factors into a product of prime ideals
• Sum of ideals A and B defined as $A + B = \{a + b | a \in A, b \in B\}$
• Product of ideals A and B defined as $AB = \{\sum a_i b_i | a_i \in A, b_i \in B, \text{finite sum}\}$
• Intersection of ideals A and B defined as $A \cap B = \{x | x \in A \text{ and } x \in B\}$
• Quotient of ideals A and B defined as $(A : B) = \{x | xB \subseteq A\}$
• Fractional ideal represents a finitely generated R-submodule of the field of fractions of R
• Inverse of ideal A defined as $A^{-1} = \{x \in K | xA \subseteq R\}$, where K denotes the field of fractions of R

Advanced Ideal Concepts

• Norm of an ideal determines its size and facilitates calculations in ideal arithmetic
  • For a number field K of degree n over Q, the norm of an ideal I is defined as $N(I) = |R/I|$, where R denotes the ring of integers of K
  • Norm is multiplicative $N(AB) = N(A)N(B)$ for ideals A and B
• Prime ideal factorization simplifies computations involving products and quotients of ideals
  • In a Dedekind domain, every non-zero ideal I can be uniquely factored as $I = P_1^{e_1} P_2^{e_2} \cdots P_k^{e_k}$, where P_i are prime ideals and e_i are positive integers
• Chinese Remainder Theorem for ideals solves systems of congruences in number fields
  • For pairwise coprime ideals I_1, I_2, ..., I_n and elements a_1, a_2, ..., a_n, there exists a unique solution modulo the product I_1 I_2 ... I_n to the system of congruences $x \equiv a_i \pmod{I_i}$

Properties of Ideal Arithmetic

Fundamental Laws of Ideal Arithmetic

• Distributive law states $A(B + C) = AB + AC$ for ideals A, B, and C
• Cancellation law asserts if $AB = AC$ and $A \neq (0)$, then $B = C$
• Associative property holds for ideal multiplication $(AB)C = A(BC)$ for ideals A, B, and C
• Commutative property applies to ideal multiplication $AB = BA$ for ideals A and B
• Identity element for ideal multiplication represents the ring R itself $AR = RA = A$ for any ideal A

Properties of Fractional Ideals

• For fractional ideals A and B, $(AB)^{-1} = A^{-1}B^{-1}$
• Set of fractional ideals of a Dedekind domain forms an abelian group under multiplication
  • Closure property holds fractional ideals A and B, AB remains a fractional ideal
  • Associativity property applies $(AB)C = A(BC)$ for fractional ideals A, B, and C
  • Identity element represents the ring R itself
  • Inverse element exists for every non-zero fractional ideal A, denoted as $A^{-1}$
• Fractional ideals allow extension of ideal arithmetic to a broader class of objects
  • Enable division of ideals in Dedekind domains
  • Facilitate the study of ideal class groups and class numbers

Ideal Operations in Examples

Computation of Ideal Operations

• Sum of ideals combines generators of each ideal and simplifies using ring properties
  • In Z[√-5], sum of ideals (2, 1 + √-5) and (3, 1 - √-5) equals (1), the entire ring
• Product of ideals multiplies generators of each ideal and simplifies using ring properties
  • In Z[i], product of ideals (2, 1 + i) and (2, 1 - i) equals (4, 2 + 2i, 2 - 2i, 1 + 2i + i^2) = (4, 1 + i)
• Quotient of ideals uses definition (A : B) to determine elements x such that xB ⊆ A
  • In Z, quotient of ideals (6) and (2) equals (3)
• Polynomial rings involve finding greatest common divisor (GCD) of polynomials for ideal arithmetic
  • In F[x], where F denotes a field, sum of ideals (x^2 + 1) and (x - 1) equals (1) if char(F) ≠ 2
• Number fields require working with algebraic integers and their minimal polynomials
  • In Q(√2), product of ideals (√2) and (3, 1 + √2) equals (3√2, √2(1 + √2))

Applications of Ideal Arithmetic

• Unique factorization of ideals into prime ideals analyzes divisibility properties
  • In Z[√-5], ideal (6) factors as (2, 1 + √-5)(2, 1 - √-5)(3, 1 + √-5)
• Ramification of primes in algebraic number fields studied through ideal arithmetic
  • Prime p ramifies in a number field K if the ideal (p) factors as $P^e$ for some prime ideal P and e > 1
• Class numbers computed using ideal arithmetic to investigate ideal class group of number field
  • Class number represents the order of the ideal class group, measuring how far the ring of integers is from being a unique factorization domain
• Dedekind zeta function connected to distribution of prime ideals through ideal arithmetic
  • Dedekind zeta function defined as $\zeta_K(s) = \sum_{I \neq 0} \frac{1}{N(I)^s}$, where sum runs over all non-zero ideals I of the ring of integers of K

Ideal Arithmetic Applications

Problem Solving with Ideal Arithmetic

• Diophantine equations in algebraic number fields solved using ideal arithmetic
  • Equation $x^2 + 5y^2 = 1$ in Z[√-5] solved by factoring (x + y√-5)(x - y√-5) = 1
• Integral bases and discriminants of number fields studied through ideal arithmetic
  • Discriminant of a number field K defined as $d_K = \det(\text{Tr}_{K/Q}(\alpha_i \alpha_j))$, where α_i form an integral basis of K
• Systems of congruences in number fields solved using Chinese Remainder Theorem for ideals
  • System $x \equiv 2 \pmod{3}, x \equiv 1 \pmod{5}$ in Z solved using CRT for ideals (3) and (5)
• Factorization and divisibility of ideals analyzed using unique prime ideal factorization
  • Ideal A divides ideal B if and only if every prime factor of A appears in the factorization of B with at least the same exponent
• Ideal class group investigated to understand structure of fractional ideals modulo principal ideals
  • Ideal class group of a number field K defined as the quotient group of fractional ideals modulo principal ideals
• Ramification indices and inertia degrees computed using ideal arithmetic in field extensions
  • For a prime ideal P of K lying above a prime p of Q, the ramification index e and inertia degree f satisfy $efg = n$, where n denotes the degree of K over Q and g denotes the number of prime ideals lying above p