Ring of Integers in a Number Field

Definition

The ring of integers in a number field is the set of algebraic integers that form a ring, which is a structure that allows for addition and multiplication. This ring consists of all elements in the number field that are roots of monic polynomials with integer coefficients, thus capturing the arithmetic properties of the number field and playing a crucial role in its structure. It connects deeply with concepts like ideals, factorization, and unique factorization domains, especially in the context of Dedekind domains.

5 Must Know Facts For Your Next Test

1. The ring of integers in a number field is always a finitely generated $b{Z}$-module, which means it can be expressed as a finite combination of its generators.
2. In the ring of integers, every non-zero prime ideal is maximal, meaning it cannot be properly contained within another ideal, reflecting the unique properties within Dedekind domains.
3. The ring of integers is integral over $b{Z}$, indicating that any element in the ring satisfies a polynomial equation with integer coefficients.
4. Factorization in the ring of integers can behave differently than in $b{Z}$; not all rings of integers exhibit unique factorization, leading to intriguing challenges in algebraic number theory.
5. The discriminant of a number field relates to the structure of its ring of integers and is crucial for understanding properties like ramification and behavior under extension.

Review Questions

• How does the concept of algebraic integers relate to the ring of integers in a number field?
  • Algebraic integers are precisely the elements that belong to the ring of integers in a number field. This relationship is foundational because algebraic integers are defined as roots of monic polynomials with integer coefficients, which inherently describes how these elements fit into the arithmetic framework of number fields. Thus, understanding algebraic integers provides insight into the structure and properties of the ring itself.
• Discuss how unique factorization fails in certain rings of integers and its implications for Dedekind domains.
  • In some rings of integers, such as those associated with non-unique factorization domains, elements may not factor uniquely into prime ideals. This contrasts with Dedekind domains, where every non-zero proper ideal factors uniquely into prime ideals. The failure of unique factorization has significant implications for solving Diophantine equations and analyzing ideal class groups within algebraic number theory.
• Evaluate the significance of discriminants in relation to rings of integers and number fields.
  • Discriminants are vital in understanding the properties of rings of integers within number fields because they provide insight into how these rings behave under various extensions. They help identify ramification issues, which can indicate how primes split or remain inert within extensions. This evaluation allows mathematicians to explore deeper aspects of arithmetic geometry and understand interactions between different algebraic structures.

Related terms

Algebraic Integers: Elements in a number field that are roots of monic polynomials with integer coefficients, which includes all the elements in the ring of integers.

Ideal: A special subset of a ring that allows for the generalization of concepts like divisibility and factorization within the structure of the ring.

Dedekind Domain: A type of integral domain where every non-zero proper ideal can be uniquely factored into prime ideals, providing a rich framework for understanding number fields and their rings of integers.