Algebraic Number

Definition

An algebraic number is a complex number that is a root of a non-zero polynomial equation with rational coefficients. This concept plays a crucial role in number theory, particularly in understanding the structure of number fields and algebraic integers. Algebraic numbers include all rational numbers and roots of polynomials, which extends the notion of integers to a broader context in mathematics.

5 Must Know Facts For Your Next Test

1. Algebraic numbers can be classified into different types based on their degree, with rational numbers being degree 1 and square roots representing degree 2.
2. All integers are considered algebraic numbers since they are roots of the polynomial equations like $$x - n = 0$$, where $$n$$ is any integer.
3. The set of algebraic numbers is countably infinite, which contrasts with the set of real numbers, which is uncountably infinite.
4. Algebraic numbers can also form field extensions, which allows for the study of their properties in higher dimensions beyond simple arithmetic.
5. Every algebraic number can be expressed in terms of its minimal polynomial, which uniquely defines it as a root while having the lowest degree possible.

Review Questions

• How do algebraic numbers relate to polynomial equations and what implications does this have for their classification?
  • Algebraic numbers are defined as roots of non-zero polynomial equations with rational coefficients. This relationship establishes various classifications of algebraic numbers based on the degree of the polynomial they satisfy. For instance, rational numbers are classified as degree 1, while roots of quadratic polynomials correspond to degree 2. Understanding this connection helps categorize different types of algebraic numbers and reveals properties inherent to their structure.
• Discuss the differences between algebraic numbers and algebraic integers, including their defining characteristics.
  • While both algebraic numbers and algebraic integers are roots of polynomials, they differ in terms of the types of polynomials they satisfy. Algebraic integers must be roots of monic polynomials with integer coefficients, whereas algebraic numbers can be roots of any polynomial with rational coefficients. This distinction means that every algebraic integer is an algebraic number, but not all algebraic numbers qualify as algebraic integers. This difference has significant implications for how we study their properties within number theory.
• Evaluate how the concept of number fields enhances our understanding of algebraic numbers and their significance in mathematics.
  • Number fields serve as an essential framework for understanding algebraic numbers by providing a structured setting in which these numbers exist and interact. By considering algebraic numbers within the context of number fields, mathematicians can explore their properties more thoroughly, including divisibility and factorization. This exploration leads to deeper insights into the structure of algebraic integers and helps unify various areas within mathematics, showcasing how abstract concepts can reveal fundamental relationships among different types of numbers.

Related terms

Algebraic Integer: An algebraic integer is a special type of algebraic number that is a root of a monic polynomial (leading coefficient is 1) with integer coefficients.

Number Field: A number field is a finite degree field extension of the rational numbers, containing algebraic numbers that can be expressed as solutions to polynomial equations.

Polynomial Equation: A polynomial equation is an expression set to zero that equates a polynomial, which consists of variables raised to non-negative integer powers multiplied by coefficients.