What is ring of o?

The ring of integers of a number field K, denoted by O_K, is a fundamental object in algebraic number theory. It consists of all elements in K that are roots of monic polynomials with integer coefficients. In other words, O_K is the integral closure of Z in K.

Here are some important aspects:

• Definition: An element α ∈ K is an algebraic integer if it is a root of a monic polynomial f(x) = xⁿ + a_n-1x^n-1 + ... + a₁x + a₀ where a_i ∈ Z for all i. O_K is the set of all algebraic integers in K.

• Ring Structure: O_K is indeed a ring, meaning it is closed under addition, subtraction, and multiplication. This can be proven rigorously.

• Examples:

  • If K = Q, then O_K = Z.
  • If K = Q(√d), where d is a square-free integer, then:
    • O_K = Z[√d] = {a + b√d | a, b ∈ Z} if d ≡ 2, 3 (mod 4).
    • O_K = Z[(1+√d)/2] = {a + b(1+√d)/2 | a, b ∈ Z} if d ≡ 1 (mod 4).

• Integral Basis: O_K is a finitely generated Z-module of rank n = [K:Q], the degree of the field extension. A basis for O_K as a Z-module is called an integral basis.

• Discriminant: Associated with an integral basis is the discriminant of K, denoted by Δ_K. The Discriminant is an important invariant that reflects arithmetic properties of the field K.

• Ideals: The study of ideals in O_K is crucial in algebraic number theory. Notably, O_K is a Dedekind domain.

• Dedekind Domain: O_K is a Dedekind%20Domain. This implies that every non-zero ideal in O_K can be uniquely factored into a product of prime ideals. This property is a generalization of the unique prime factorization of integers in Z.

• Units: The group of units O_K^× consists of the invertible elements in O_K. Dirichlet's unit theorem describes the structure of this group.

• Applications: The ring of integers plays a vital role in various areas of number theory, including the study of Diophantine equations, class field theory, and cryptography.