An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. These numbers include all integers and roots of unity, forming a crucial part of algebraic number theory, particularly when discussing the properties of number fields, norms, traces, minimal polynomials, and the structure of Dedekind domains.
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Algebraic integers are closed under addition and multiplication, meaning that the sum or product of any two algebraic integers is also an algebraic integer.
The set of algebraic integers forms an integral domain, which means it has no zero divisors and allows for division with remainders.
For any algebraic integer, its norm (the product of its conjugates) is an integer, providing insight into the structure of algebraic integers in number fields.
The trace of an algebraic integer (the sum of its conjugates) can also be used to characterize its properties and determine relationships within its minimal polynomial.
Every integer is an algebraic integer, but not every algebraic integer is an ordinary integer; some can be roots of polynomials with complex coefficients.
Review Questions
How do algebraic integers relate to number fields and what properties do they exhibit?
Algebraic integers are essential components within number fields, as they serve as roots of monic polynomials with integer coefficients. They share important properties such as closure under addition and multiplication, making them integral domains. The study of norms and traces associated with these integers reveals deeper insights into their relationships within the number field.
Discuss the significance of minimal polynomials in understanding algebraic integers.
Minimal polynomials play a critical role in characterizing algebraic integers by providing the monic polynomial of least degree that has the integer as a root. These polynomials are irreducible over the integers, allowing us to determine properties like the norm and trace. The roots of these minimal polynomials help identify how algebraic integers interact within their respective number fields.
Evaluate how the concept of Dedekind domains enhances our understanding of algebraic integers and their applications in number theory.
Dedekind domains enrich our understanding of algebraic integers by providing a framework where every non-zero prime ideal is maximal. This structure helps in analyzing the factorization properties of algebraic integers, including unique factorization and divisibility. Furthermore, studying algebraic integers within Dedekind domains allows mathematicians to explore relationships between different ideals, leading to deeper insights into the arithmetic properties and behaviors of these numbers in various number theoretic contexts.
Related terms
Number Field: A number field is a finite degree field extension of the rational numbers, which allows for the study of algebraic integers within a broader context.
Minimal Polynomial: The minimal polynomial of an algebraic integer is the monic polynomial of least degree that has the integer as a root and is irreducible over the integers.
Dedekind Domain: A Dedekind domain is a special type of integral domain where every non-zero prime ideal is maximal, which facilitates the understanding of algebraic integers and their properties.