Algebraic Integers

Definition

Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients, making them a key component in the study of number theory and algebraic number fields. These integers generalize the notion of regular integers and play a significant role in various mathematical structures, including unique factorization in Dedekind domains, properties related to units, and connections to zeta functions.

5 Must Know Facts For Your Next Test

1. Every algebraic integer is also an algebraic number, but not all algebraic numbers are algebraic integers.
2. The ring of algebraic integers in a number field is a Dedekind domain, which means it has nice properties regarding the factorization of ideals.
3. The existence of units among algebraic integers helps to determine the structure of the class group in algebraic number theory.
4. Complex multiplication theories often utilize algebraic integers to study elliptic curves and their properties over different fields.
5. The Dedekind zeta function is a crucial tool for understanding the distribution of prime ideals in the context of algebraic integers.

Review Questions

• How do algebraic integers relate to Dedekind domains, and why is this relationship significant?
  • Algebraic integers are integral elements in a Dedekind domain, which means they have well-defined factorization properties. The significance lies in the ability to uniquely factor non-zero proper ideals into prime ideals in Dedekind domains. This unique factorization mirrors the unique factorization of integers into primes, allowing mathematicians to extend results from classical number theory into the realm of algebraic numbers.
• Discuss how the properties of units among algebraic integers influence the structure of the class group.
  • Units among algebraic integers contribute significantly to the structure of the class group, which measures the failure of unique factorization in rings of integers. The class group is formed by equivalence classes of fractional ideals, and units help define the relation between these classes. By identifying units, we can better understand the relationships within the class group, revealing important information about the overall structure and behavior of algebraic integers.
• Evaluate the role of algebraic integers in defining and studying Dedekind zeta functions and their implications for number theory.
  • Algebraic integers play a pivotal role in defining Dedekind zeta functions, which are generating functions that encode information about prime ideals in number fields. The properties and distribution of these prime ideals are intimately connected to the structure of algebraic integers. Analyzing Dedekind zeta functions allows mathematicians to derive results related to class numbers and other important aspects of number theory, ultimately enhancing our understanding of how primes behave in different algebraic settings.

Related terms

Algebraic Number Field: A field extension of the rational numbers that is finite-dimensional and can be generated by adjoining a root of a polynomial with rational coefficients.

Dedekind Domain: A special type of integral domain where every non-zero proper ideal can be uniquely factored into prime ideals, providing a framework for studying algebraic integers.

Units: Elements in a ring that have multiplicative inverses, which are important when exploring the structure of algebraic integers and their properties.