Algebraic Number
Definition
An algebraic number is a complex number that is root of a non-zero polynomial equation with rational coefficients. In other words, it is a complex number that can be expressed as the root of an equation of the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0\), where all the coefficients are rational numbers and \(a_n \neq 0\).
History
The concept of algebraic numbers dates back to the 17th century, when it was first introduced by the Italian mathematician Bernhard Riemann. He showed that the set of algebraic numbers is closed under addition, subtraction, multiplication, division, and taking roots. However, he also noted that not all elements of the set are rational, which led him to introduce Complex numbers.
Characterization
An algebraic number can be characterized as follows:
- It must satisfy an equation with rational coefficients.
- The equation is irreducible over the rationals, meaning it cannot be factored into polynomials with rational coefficients.
- The number must not be a root of any polynomial equation with integer coefficients.
Types of Algebraic Numbers
There are several types of algebraic numbers:
- Rational Algebraic Numbers: These are Complex numbers that can be expressed as the root of an equation with rational coefficients. Examples include the square root of 2 and the cube root of 3.
- Transcendental Algebraic Numbers: These are Complex numbers that cannot be expressed as a root of any polynomial equation with rational coefficients. Examples include the imaginary unit i (the square root of -1).
- Algebraic Irrational Numbers: These are Complex numbers that can be expressed as the root of an equation with rational coefficients, but not all elements of the set are transcendental. Examples include π and e.
Properties
Some important properties of algebraic numbers include:
- Unique Factorization: Algebraic numbers always have a unique factorization into prime factors.
- Galois Theory: The study of Galois groups is an important area of study in Algebraic Number Theory, which provides insights into the behavior of polynomials over finite fields.
Applications
Algebraic numbers have numerous applications in mathematics and computer science:
- Cryptography: Algebraic numbers play a crucial role in cryptography, particularly in public-key cryptography.
- Number Theory: Algebraic numbers are used to study number theory topics such as primality, divisibility, and Modular forms.
- Computer Science: Algebraic numbers are used in computer science to optimize algorithms for tasks such as computing greatest common divisors and prime factorizations.
Notable Theorems
Some important theorems in Algebraic Number Theory include:
- Riemann Hypothesis: This theorem, proposed by Bernhard Riemann in 1859, states that all non-trivial zeros of the Riemann zeta function are algebraic numbers.
- Modular forms: These are complex-valued functions that are defined on the upper half-plane and satisfy certain modular properties. They have applications in number theory and geometry.
Notable Researchers
Some notable researchers in Algebraic Number Theory include:
- Bernhard Riemann: A German mathematician who introduced the concept of algebraic numbers and studied their properties.
- David Hilbert: A German mathematician who made significant contributions to number theory, including the study of congruences and the classification of elliptic curves.
Sources
For further reading on algebraic numbers, consider the following sources:
- “Algebraic Numbers” by Richard Taylor
- “Number Theory: An Introduction to the Ideas and Methods of Modern Number Theory” by Henryk Iwaniec and Emmanuel Kowalski
- “Real and Complex Analysis” by G. David Birkhoff