Algebraic Number Field

Definition

An algebraic number field is a finite extension of the field of rational numbers, meaning it can be constructed by adjoining roots of polynomial equations with coefficients in the rationals. This concept is fundamental in number theory, as it allows for a deeper understanding of number systems that include algebraic integers, which are roots of monic polynomials with integer coefficients. Properties such as unique factorization and the behavior of units can be studied within these fields, connecting them to broader topics like unit groups and class groups.

5 Must Know Facts For Your Next Test

1. Algebraic number fields can be classified as either Galois or non-Galois, depending on whether every irreducible polynomial splits completely into linear factors over the field.
2. Every algebraic number field contains a set of algebraic integers, which are crucial for studying Diophantine equations and other number-theoretic questions.
3. The ring of integers of an algebraic number field is a Dedekind domain, which ensures unique factorization of ideals into prime ideals.
4. In any algebraic number field, the units can be understood through Dirichlet's unit theorem, which characterizes the group of units in terms of the field's rank and regulators.
5. Algebraic number fields play a key role in understanding class numbers, which measure the failure of unique factorization within the ring of integers.

Review Questions

• How does the concept of an algebraic number field extend our understanding of rational numbers?
  • An algebraic number field extends our understanding of rational numbers by including roots of polynomial equations that are not necessarily rational themselves. For example, when we adjoin $ ext{sqrt}(2)$ to $ ext{Q}$, we create an algebraic number field that allows us to work with numbers that can't be expressed as simple fractions. This inclusion leads to exploring properties such as divisibility and factorization in a more complex environment.
• Discuss how Dirichlet's unit theorem applies to algebraic number fields and its significance in number theory.
  • Dirichlet's unit theorem provides valuable insight into the structure of the group of units in an algebraic number field. It states that this group is finitely generated and can be expressed as a product of a free abelian group and a finite cyclic group. This understanding is significant because it helps us determine the nature of solutions to various Diophantine equations and establishes connections between different fields in number theory.
• Evaluate the implications of class numbers in algebraic number fields for unique factorization.
  • The class number in an algebraic number field measures how far from unique factorization its ring of integers strays. If the class number is one, it indicates that every ideal factors uniquely into prime ideals, reinforcing unique factorization. However, if the class number is greater than one, it implies that unique factorization fails, leading to intricate relationships among prime elements and impacting many areas within both algebra and number theory.

Related terms

Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients.

Degree of a Field Extension: The degree of a field extension is the dimension of the extended field as a vector space over the base field, indicating how many times larger the extension is compared to the base field.

Norm: The norm of an element in a number field is the product of its conjugates, providing important information about the element's properties and its relationship with the field.