Number Field

Definition

A number field is a finite degree extension of the rational numbers, $ extbf{Q}$, which means it contains elements that can be expressed as roots of polynomials with coefficients in $ extbf{Q}$. This concept is crucial as it allows the study of arithmetic properties through the lens of algebraic structures, linking to ideals, units, and class groups in a systematic way. Number fields serve as the foundation for understanding the behavior of algebraic integers, the distribution of primes, and extensions related to local and global fields.

5 Must Know Facts For Your Next Test

1. Every number field has an associated ring of integers, which consists of all algebraic integers within that field.
2. The degree of a number field over $ extbf{Q}$ provides insights into its structure and properties, such as its Galois group when applicable.
3. Number fields can be classified based on their discriminants, which help in understanding their arithmetic behavior and potential applications.
4. The units in the ring of integers of a number field relate directly to Dirichlet's unit theorem, describing how these units form a finitely generated group.
5. Hilbert's class field theory connects number fields with abelian extensions, allowing for an exploration of Galois groups and class groups.

Review Questions

• How do the properties of a number field impact its ring of integers and their ideals?
  • The properties of a number field significantly influence its ring of integers by determining the nature of its ideals. In particular, if the number field is a Dedekind domain, every non-zero ideal can be uniquely factored into prime ideals. This structure facilitates the study of both class groups and unit groups within that field, allowing us to better understand how these concepts interact and contribute to the overall arithmetic properties.
• Discuss the relationship between units in a number field and Dirichlet's unit theorem.
  • Units in the ring of integers of a number field form a finitely generated group, which is described by Dirichlet's unit theorem. This theorem states that the group of units can be decomposed into two parts: a finite subgroup related to roots of unity and a free part whose rank is determined by the number of real embeddings minus the number of complex embeddings. This relationship helps in understanding how units behave under multiplication and their role in solving Diophantine equations.
• Evaluate how Hilbert class fields provide insights into the ideal class group of a number field.
  • Hilbert class fields are significant because they serve as the maximal unramified abelian extension of a given number field. They illuminate the structure of the ideal class group by establishing a correspondence between ideal classes and certain Galois groups associated with these fields. This relationship enhances our understanding by illustrating how class field theory connects local properties with global behaviors in number fields, ultimately revealing deeper connections among various algebraic structures.

Related terms

Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients, playing a key role in the structure of number fields.

Dedekind Domain:

A Dedekind domain is a special type of integral domain where every nonzero prime ideal is maximal, providing a framework for studying number fields and their ring of integers.

Ideal Class Group:

The ideal class group is a group that measures the failure of unique factorization in the ring of integers of a number field, indicating how ideals behave in that field.