Algebraic Integers

Definition

Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They play a crucial role in number theory, particularly in the study of unique factorization and properties of number fields, connecting various concepts like integral bases and prime ideals.

5 Must Know Facts For Your Next Test

1. Algebraic integers include all integers and can also include roots of polynomials like $$x^2 + x + 1$$ which has roots that are not rational.
2. The set of algebraic integers forms a ring, meaning they are closed under addition and multiplication, leading to interesting properties in number theory.
3. Not all algebraic integers are prime; some may factor further into other algebraic integers within their respective rings.
4. In many algebraic number fields, the presence of non-prime algebraic integers leads to unique factorizations being complicated or failing altogether.
5. Algebraic integers are closely linked to class groups, as their properties influence the structure and finiteness of such groups in algebraic number theory.

Review Questions

• How do algebraic integers relate to the concept of unique factorization in number theory?
  • Algebraic integers are essential in understanding unique factorization because they are the building blocks for rings of integers in various number fields. While many rings allow for unique factorization, some do not due to the presence of non-prime algebraic integers that can complicate this property. The study of unique factorization leads to the investigation of whether specific rings are Unique Factorization Domains (UFDs), impacting how we analyze numbers within those fields.
• Discuss the importance of algebraic integers in defining the ring of integers in a number field and how it impacts number theory.
  • The ring of integers in a number field is comprised entirely of algebraic integers, which directly influences its structure and properties. These rings form the foundation for much of algebraic number theory, as they allow for the application of arithmetic concepts to non-rational numbers. Understanding these rings helps mathematicians explore more complex topics such as ideal class groups and their relationships with prime ideals within these fields.
• Evaluate how the existence and properties of algebraic integers affect the finiteness of class numbers in algebraic number fields.
  • The existence and properties of algebraic integers play a pivotal role in determining the finiteness of class numbers within algebraic number fields. Class numbers measure how far a given ring deviates from unique factorization; when rings contain many algebraic integers with complex relationships, this can lead to larger or infinite class numbers. The analysis involves intricate connections between norms, traces, and ideal classes, showcasing how algebraic integers directly influence the underlying arithmetic structure of these fields.

Related terms

Monic Polynomial: A polynomial where the leading coefficient (the coefficient of the highest power) is equal to 1.

Ring of Integers: The set of algebraic integers in a number field, forming a ring under standard addition and multiplication.

Unique Factorization Domain (UFD): A type of integral domain in which every non-zero element can be factored uniquely into irreducible elements, up to order and units.