Algebraic numbers

Algebraic numbers are a subset of complex numbers that are roots of non-zero polynomial equations with rational coefficients. In other words, a complex number \( \alpha \) is considered algebraic if there exists a polynomial \( P(x) \) with \( P(x) \in \mathbb{Q}[x] \) (the set of all polynomials with rational coefficients) such that \( P(\alpha) = 0 \).

Cubic irrational numbers are numbers that can be expressed as the root of a cubic polynomial with rational coefficients, and they are not expressible as a fraction of two integers.

Quadratic irrational numbers are a type of irrational number that can be expressed in the form \( \frac{a + b\sqrt{d}}{c} \), where \( a \), \( b \), and \( c \) are integers, \( d \) is a non-square positive integer, and \( c \) is a positive integer. In simpler terms, they can be represented as a root of a quadratic equation with integer coefficients.

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. In mathematical terms, a rational number can be represented as: \[ \frac{a}{b} \] where \( a \) and \( b \) are integers, and \( b \neq 0 \).

An **algebraic integer** is a special type of number in algebraic number theory. It is defined as a complex number that is a root of a monic polynomial (a polynomial whose leading coefficient is 1) with integer coefficients.

An **algebraic number** is a complex number that is a root of a non-zero polynomial equation with rational coefficients.

A **constructible number** is a number that can be constructed using a finite number of operations involving basic geometric tools: a compass and a straightedge. This means that the number can be represented through a series of steps including drawing straight lines, constructing circles, and finding points of intersection, starting from a point set at a distance of one unit.

Eisenstein integers are a special type of complex numbers that can be expressed in the form: \[ z = a + b\omega \] where \( a \) and \( b \) are integers, and \( \omega \) is a primitive cube root of unity.

A **Gaussian integer** is a complex number of the form \( a + bi \), where \( a \) and \( b \) are both integers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In other words, Gaussian integers are the set of complex numbers whose real and imaginary parts are both whole numbers.

Geometric constructions are methods used to create geometric figures or shapes using only a compass and a straightedge, without any measurements. This involves combining points, lines, and circles to arrive at desired geometric figures based on certain rules and principles of geometry. The fundamental tools of geometric construction are: 1. **Straightedge**: A tool used to draw straight lines between two points. It cannot be used to measure distances or for marking specific lengths.

The imaginary unit, denoted as \( i \), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\). This means: \[ i^2 = -1 \] The imaginary unit allows for the extension of the number system to include numbers that cannot be represented on the traditional number line.

A Perron number is a specific type of algebraic integer that is a root of a polynomial with integer coefficients and has certain distinct properties. Specifically, a Perron number is defined as an algebraic integer \(\alpha\) that is greater than 1 and satisfies the condition that: 1. The conjugates of \(\alpha\) (all the roots of its minimal polynomial) are all less than or equal to \(\alpha\).

A Pisot–Vijayaraghavan (PV) number is a type of algebraic number that is a real root of a monic polynomial with integer coefficients, where this root is greater than 1, and all other roots of the polynomial, which can be real or complex, lie inside the unit circle in the complex plane (i.e., have an absolute value less than 1).

Roth's theorem is a result in number theory that pertains to the distribution of arithmetic progressions in subsets of natural numbers. It is particularly significant in additive combinatorics and deals with the existence of long arithmetic progressions within sets of integers. The theorem states that any subset \( A \) of the integers (specifically, the natural numbers) with positive upper density cannot avoid having an arithmetic progression of length 3.

A Salem number is a type of algebraic integer that is defined as a root of a polynomial with integer coefficients, where the polynomial has a degree of at least 2, and at least one of its roots lies outside the unit circle in the complex plane. Specifically, a Salem number is a real number greater than 1, and all of its other Galois conjugates (roots) are located inside or on the unit circle.