Ring of integers of a number field

Definition

The ring of integers of a number field is the integral closure of the integers in that field, serving as a generalization of the concept of integers to more complex algebraic structures. This ring consists of all elements in the number field that are roots of monic polynomials with coefficients in the integers, forming a critical structure for understanding the arithmetic properties of number fields. The ring of integers plays a key role in defining Dedekind domains, which are integral domains where every non-zero prime ideal is maximal.

5 Must Know Facts For Your Next Test

1. The ring of integers is not just the usual integers but can include numbers like $$\frac{1 + \sqrt{-5}}{2}$$ when considering certain number fields.
2. The ring of integers of a number field has unique factorization properties similar to those in the ring of integers, meaning every non-zero element can be factored into irreducible elements.
3. The class group, which measures the failure of unique factorization in the ring of integers, plays an important role in understanding the structure and properties of the number field.
4. For every number field, its ring of integers can be constructed using minimal polynomials over the rationals, allowing for systematic exploration of its elements.
5. The discriminant of a number field gives insight into the structure and complexity of its ring of integers and helps determine whether it is a Dedekind domain.

Review Questions

• How does the ring of integers relate to unique factorization and what implications does this have for algebraic integers?
  • The ring of integers serves as an essential framework for understanding unique factorization within algebraic integers. In this ring, every non-zero element can typically be uniquely factored into irreducible elements. However, there may be exceptions in some number fields where unique factorization fails, leading to the concept of the class group. This relationship highlights how rings can exhibit both similar and unique properties compared to traditional integer factorization.
• Discuss how the concept of Dedekind domains connects to the structure and properties of the ring of integers in number fields.
  • Dedekind domains are defined by their property that every non-zero prime ideal is maximal, a characteristic deeply linked to the ring of integers in number fields. The ring of integers naturally forms a Dedekind domain when it possesses unique factorization. This connection allows mathematicians to apply properties from Dedekind domains, such as ideal class groups and localizations, to gain insights into the arithmetic behavior within number fields and their rings.
• Evaluate the importance of the discriminant in understanding the properties of rings of integers within number fields and how it affects their classification.
  • The discriminant serves as a pivotal tool for categorizing and analyzing rings of integers in number fields. It encodes essential information regarding the roots' relationships and their algebraic dependencies. Specifically, it helps determine whether a ring is a Dedekind domain by indicating potential factorization complexities. Additionally, its value can signal how 'wild' or 'tame' certain behaviors are within the algebraic structure, influencing both theoretical explorations and practical applications in algebraic number theory.

Related terms

Number Field: A number field is a finite field extension of the rational numbers, consisting of all numbers that can be expressed as roots of polynomial equations with rational coefficients.

Dedekind Domain: A Dedekind domain is an integral domain in which every non-zero proper ideal factors uniquely into prime ideals, which is essential for the study of algebraic integers.

Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients, and these numbers form the elements of the ring of integers in a number field.