Field (mathematics)

In mathematics, a **field** is a set equipped with two binary operations that generalize the arithmetic of rational numbers. These operations are typically called addition and multiplication, and they must satisfy certain properties. Specifically, a field is defined as follows: 1. **Closure**: For any two elements \( a \) and \( b \) in the field, both \( a + b \) and \( a \cdot b \) are also in the field.

Algebraic number theory is a branch of mathematics that studies the properties of numbers in the context of algebraic structures, particularly focusing on the algebraic properties of integers, rational numbers, and their extensions. It combines elements of both number theory and abstract algebra, particularly through the study of number fields and their rings of integers. Key concepts in algebraic number theory include: 1. **Number Fields**: These are finite degree extensions of the field of rational numbers (ℚ).

Class field theory is a branch of algebraic number theory that explores the connections between number fields and their algebraic structure through the lens of Galois theory. It primarily aims to study abelian extensions of number fields, which are extensions of number fields that are Galois with an abelian Galois group. The theory provides a correspondence between the ideals of a number field and the abelian extensions of that field.

A field extension is a fundamental concept in abstract algebra, specifically in the study of fields. A field is a set equipped with two operations (usually called addition and multiplication) that satisfy certain axioms, including the existence of multiplicative and additive inverses. A field extension is essentially a larger field that contains a smaller field as a subfield.

Finite fields, also known as Galois fields, are algebraic structures that consist of a finite number of elements and possess operations of addition, subtraction, multiplication, and division (excluding division by zero) that satisfy the field properties. A field is defined by the following properties: 1. **Closure**: The set is closed under the operations of addition, subtraction, multiplication, and non-zero division. 2. **Associativity**: Both addition and multiplication are associative.

Galois theory is a branch of abstract algebra that studies the relationships between field extensions and group theory, particularly focusing on the solvability of polynomial equations. Named after the mathematician Évariste Galois, it provides a powerful framework for understanding how the roots of polynomials are related to the symmetry properties of the equations. The core ideas of Galois theory can be summarized as follows: 1. **Field Extensions**: A field extension is a bigger field that contains a smaller field.

An **algebraic function field** is a type of mathematical structure that serves as a generalization of both algebraic number fields and function fields over finite fields.

An **algebraic number field** is a certain type of field in algebraic number theory. Specifically, an algebraic number field is a finite extension of the field of rational numbers, \(\mathbb{Q}\), that is generated by the roots of polynomial equations with coefficients in \(\mathbb{Q}\).

An **algebraically closed field** is a field \( F \) in which every non-constant polynomial equation with coefficients in \( F \) has at least one root in \( F \).

An "all one polynomial" typically refers to a polynomial where every coefficient is equal to one.

The Archimedean property is a fundamental concept in mathematics that relates to the behavior of real numbers, particularly in the context of the ordering of numbers. It states that for any two positive real numbers \( a \) and \( b \), there exists a natural number \( n \) such that: \[ n \cdot a > b.

The Brauer–Wall group is an important concept in the field of algebra, particularly in algebraic K-theory and the theory of central simple algebras. It is named after mathematicians Richard Brauer and Norman Wall. ### Definition The Brauer–Wall group, often denoted \( Br(W) \), is defined in relation to a given ring \( R \).

A CM-field, short for "Complex Multiplication field," is a type of number field that is significant in algebraic number theory, particularly in the study of elliptic curves and modular forms. More specifically, a CM-field is an imaginary quadratic field \(K\) that arises from the theory of elliptic curves with complex multiplication by a certain ring of integers.

In algebra, particularly in the context of field theory and ring theory, the characteristic of a ring or field is a fundamental concept that essentially describes how many times you can add the identity element to itself before reaching the additive identity (zero).

In field theory, particularly in the context of abstract algebra and number theory, the concept of a "conjugate element" often refers to the behavior of roots of polynomials and their extensions in fields. ### Conjugate Elements in Field Theory 1. **Field Extensions**: When we have a field extension \( K \subset L \), elements of \( L \) that are roots of a polynomial with coefficients in \( K \) are called conjugates of each other.

A cubic field is a specific type of number field, which is a finite field extension of the rational numbers \(\mathbb{Q}\) of degree three. In more formal terms, a cubic field is generated by extending \(\mathbb{Q}\) with an element \(\alpha\) such that the minimal polynomial of \(\alpha\) over \(\mathbb{Q}\) is a polynomial of degree three.

A **discrete valuation** is a special type of valuation defined on a field, which gives a way to measure the "size" of elements in that field. More specifically, a discrete valuation provides a way to assess how "close" elements are to zero in a field, often in the context of algebraic number theory or local fields.

Eisenstein's criterion is a useful test for determining the irreducibility of a polynomial with integer coefficients over the field of rational numbers (or equivalently, over the integers). It is named after the mathematician Gotthold Eisenstein.

An equally spaced polynomial, also known as a polynomial interpolating at equally spaced nodes, is a type of polynomial that passes through a set of points (nodes) that are spaced evenly on the x-axis. This concept is often used in numerical analysis, particularly in polynomial interpolation.

The term "Euclidean field" can refer to several concepts depending on the context in mathematics and physics, but it isn't a widely recognized term on its own. Here are a couple of interpretations: 1. **In Mathematics**: A Euclidean field might refer to a field that is equipped with a Euclidean metric (or distance function) that satisfies the properties of a Euclidean space.

The **field of fractions** is a concept in algebra that deals with the construction of a field from an integral domain. An integral domain is a type of commutative ring with no zero divisors and a unity (1 ≠ 0). The field of fractions allows us to create a field in which the elements can be expressed as fractions (ratios) of elements from the integral domain.

Field trace can refer to different concepts depending on the context, so I'll outline a few possible interpretations: 1. **General Definition**: In a broad sense, a field trace could refer to a record or representation of observations or data collected from a specific field or area of study. This could be used in various disciplines, such as ecology, geography, or even data science.

A formally real field is a type of field in mathematics that adheres to certain properties regarding sums of squares. Specifically, a field \( K \) is said to be formally real if it does not contain any non-negative elements that cannot be expressed as a sum of squares of elements from \( K \).

The Function Field Sieve (FFS) is an algorithm used for factoring large integers, particularly those that are difficult to factor with classical methods. It extends the ideas of the number field sieve (NFS), which is currently one of the most efficient known methods for factoring large composite numbers, especially those with large prime factors.

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree \( n \) with complex coefficients has exactly \( n \) roots in the complex number system, counting multiplicities.

A generic polynomial is a polynomial that is defined with coefficients that can represent any number, typically treated as indeterminate or symbolic variables.

The term "global field" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **In Mathematics (Field Theory)**: In mathematics, particularly in algebra, a global field is a specific type of field that is either a number field (a finite field extension of the rational numbers) or a function field over a finite field (a field of rational functions in one variable over a finite field).

A glossary of field theory typically consists of key terms and concepts related to the study of field theory, which is a fundamental area in physics and mathematics, particularly in the realms of quantum mechanics, particle physics, and general relativity. Here are some common terms you might find in a glossary of field theory: 1. **Field**: A physical quantity represented at every point in space and time, such as an electromagnetic field or gravitational field.

In mathematics, specifically in algebra, a "ground field" (often simply referred to as a "field") is a basic field that serves as the foundational set of scalars for vector spaces and algebraic structures.

The Hasse invariant is a fundamental concept in the theory of algebraic forms and is particularly important in the study of quadratic forms over fields, especially in relation to the classification of these forms under certain equivalences. Given a finite-dimensional algebra over a field, the Hasse invariant provides a way to distinguish between different algebraic structures.

The term "higher local field" typically refers to specific types of fields in algebraic number theory, particularly in relation to local fields and their extensions. In this context, local fields are complete fields with respect to a discrete valuation, which often arise in number theory. Common examples include the field of p-adic numbers and complete extensions of the rational numbers.

The Hurwitz problem, named after the mathematician Adolf Hurwitz, concerns the enumeration of the ways to express a given integer as a sum of two or more squares. Specifically, it explores questions related to which integers can be represented as sums of squares and the number of distinct ways in which a number can be expressed as such.

Hyperreal numbers are an extension of the real numbers that include infinitesimal and infinite quantities. They are used in non-standard analysis, a branch of mathematics that reformulates calculus and analysis using these quantities. The hyperreal number system is constructed by taking sequences of real numbers and using an equivalence relation to group them.

Iwasawa theory is a branch of number theory that studies the properties of number fields and their associated Galois groups using techniques from algebraic geometry, modular forms, and the theory of L-functions. Named after the Japanese mathematician K. Iwasawa, the theory primarily focuses on the arithmetic of cyclotomic fields and \( p \)-adic numbers, and it aims to understand the behavior of various arithmetic objects in relation to these fields.

The Jacobson–Bourbaki theorem is a result in the field of algebra, specifically in the theory of rings and algebras. It provides a characterization of the Jacobson radical of a ring in terms of the ideal structure of that ring. The theorem can be stated as follows: Let \( R \) be a commutative ring with unity, and let \( \mathfrak{m} \) be a maximal ideal of \( R \).

Krasner's lemma is a result in the field of number theory, specifically dealing with linear forms in logarithms of algebraic numbers. It provides conditions under which a certain linear combination of logarithms can lead to a rational approximation or a specific form of representation. The lemma is often used in Diophantine approximation and transcendency theory.

Kummer theory, named after the mathematician Ernst Eduard Kummer, is a branch of number theory that deals with the study of the behavior of prime numbers in relation to fields and their extensions, particularly focusing on certain types of algebraic numbers known as "Kummer extensions." Here are the key points related to Kummer theory: 1. **Kummer Extensions**: These are specific extensions of number fields obtained by adjoining roots of elements.

The term "Levi-Civita field" does not correspond to a well-defined concept widely recognized in mathematics or physics. However, it seems like you might be referring to a couple of distinct but related concepts: the Levi-Civita symbol (or tensor) and the Levi-Civita connection in the context of differential geometry.

The term "linked field" can refer to different concepts depending on the context. Here are a couple of interpretations: 1. **Database Context**: In databases, a linked field might refer to a field in a database table that is connected to a field in another table. This is often part of a relational database design, where relationships between tables are established through foreign keys.

Liouville's theorem in differential algebra concerns the conditions under which certain differential equations can be integrated in terms of elementary functions.

A number field is a finite degree extension of the field of rational numbers \(\mathbb{Q}\). The class number of a number field is an important invariant that measures the failure of unique factorization in its ring of integers. A number field with class number one has unique factorization, which is a desirable property in algebraic number theory.

The term "local field" can refer to different concepts in different contexts, including mathematics, physics, and other fields. Here are two common meanings: 1. **Local Fields in Number Theory**: In the context of algebraic number theory, a local field is a complete field with respect to a discrete valuation, which is often associated with the study of numbers in number fields. These fields are typically used to examine the local properties of arithmetic objects.

Lüroth's theorem is a result in the field of algebraic geometry and number theory, specifically concerning the field of rational functions. It states that if \( K \) is a field of characteristic zero, any finitely generated field extension \( L/K \) that is purely transcendental (i.e.

In field theory, the minimal polynomial of an element \(\alpha\) over a field \(F\) is the monic polynomial of least degree with coefficients in \(F\) that has \(\alpha\) as a root. More specifically, the minimal polynomial has the following properties: 1. **Monic**: The leading coefficient (the coefficient of the highest degree term) is equal to 1.

Nagata's conjecture is a statement in the field of algebraic geometry, particularly concerning algebraic varieties in projective space. Specifically, it pertains to the relationships between the dimensions of varieties and the degrees of their defining equations.

"Norm form" can refer to different concepts depending on the context, such as mathematics, particularly in linear algebra and functional analysis, or abstract algebra. Here are a couple of interpretations: 1. **Norm in Linear Algebra**: In the context of linear algebra, a norm represents a function that assigns a non-negative length or size to vectors in a vector space.

P-adic numbers are a system of numbers used in number theory that extend the classical notion of integers and rationals to include a different form of "closeness" or convergence. The term "p-adic" refers to a prime number \( p \), and the concept is based on an alternative metric or valuation defined by \( p \).

A **p-adically closed field** is a field that satisfies certain properties related to valuation theory and algebraic closure in the context of p-adic numbers. To understand it fully, let's break it down: 1. **p-adic Numbers**: The p-adic numbers \( \mathbb{Q}_p \) are a system of numbers used in number theory.

In the context of differential geometry and algebraic geometry, a **P-basis** typically refers to a basis for a vector space that is relevant to a particular property or structure denoted by "P." The term can have different meanings depending on the specific field or application; for instance: 1. **In Linear Algebra**: A P-basis could refer to a basis of a module or vector space that fulfills certain properties defined by "P.

A *perfect field* is a specific type of field in abstract algebra that has certain desirable properties, particularly in relation to algebraic extensions and the behavior of polynomials.

In field theory, a **primitive polynomial** is a special type of polynomial that plays a significant role in constructing finite fields (also known as Galois fields) and in various areas of algebra.

A pseudo-finite field is a structure that has properties resembling those of finite fields but is not actually finite itself. Specifically, it is an infinite field that behaves like a finite field in various algebraic respects.

A pseudo-algebraically closed field is a concept from field theory, particularly in the area of model theory and algebraic geometry. It is a type of field that can be seen as a generalization of algebraically closed fields, but without all the restrictive properties of a complete algebraic closure.

In the context of field theory in mathematics, a purely inseparable extension is a type of field extension that arises primarily in the study of fields of positive characteristic, particularly finite fields and their extensions.

The term "Pythagorean number" commonly refers to the values (typically integers) that can be the lengths of the sides of a right triangle when following the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

A Pythagorean field is a specific type of field in mathematics that is characterized by the property that every non-zero element in the field is a sum of two squares.

A quadratic field is a specific type of number field that is generated by adjoining a square root of a rational number to the field of rational numbers, \(\mathbb{Q}\). More formally, a quadratic field can be expressed in the form: \[ K = \mathbb{Q}(\sqrt{d}) \] where \(d\) is a square-free integer (an integer not divisible by a perfect square greater than 1).

A **quadratically closed field** is a type of field in which every non-constant polynomial of degree two has a root.

A quasi-algebraically closed field is a concept from field theory, specifically in the area of algebraic geometry and model theory. A field \( K \) is said to be quasi-algebraically closed if every non-constant polynomial in one variable, when considered over \( K \), has a root in the algebraic closure of \( K \).

A quasi-finite field is a concept primarily encountered in the context of algebra and field theory. However, the term is not widely used, and you might be referring to a specific aspect of finite fields or a field theory construct. In general terms, a finite field (also called a Galois field) is a field that contains a finite number of elements. Finite fields are well-studied in mathematics, particularly in number theory, coding theory, and algebraic geometry.

A quasifield is a mathematical structure that generalizes the concept of a field. In particular, a quasifield is a set equipped with two binary operations (often referred to as addition and multiplication) that satisfy certain axioms resembling those of a field, but with some modifications. In a quasifield, the operations are defined in a way that allows for the existence of division (except by zero), meaning that every nonzero element has a multiplicative inverse.

A quaternionic structure refers to a mathematical framework or system that originates from the quaternions, which are a number system that extends complex numbers.

A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not equal to zero. In other words, rational numbers include integers, finite decimals, and repeating decimals. For example: - The number \( \frac{1}{2} \) is a rational number.

In algebraic geometry, a **rational variety** is a type of algebraic variety that has a non-constant rational function defined on it that is, in some sense, "simple" or "well-behaved.

A **real closed field** is a type of field in which certain algebraic properties analogous to those of the real numbers hold. More formally, a field \( K \) is called a real closed field if it satisfies the following conditions: 1. **Algebraically Closed**: Every non-constant polynomial in one variable with coefficients in \( K \) has a root in \( K \).

The term "rupture field" can refer to different concepts depending on the context, particularly in fields like geology, seismology, or even in social sciences. Below are a couple of contexts where "rupture field" might be relevant: 1. **Geology/Seismology**: In the context of tectonic plates and earthquake studies, a "rupture field" often refers to the area affected by the rupture of a fault during an earthquake.

A separable polynomial is a polynomial that does not have repeated roots in its splitting field. More formally, a polynomial \( f(x) \) over a field \( K \) is termed separable if its derivative \( f'(x) \) and \( f(x) \) share no common roots in an algebraic closure of \( K \).

Serre's Conjecture II pertains to the field of algebraic geometry and representation theory, specifically concerning the properties of vector bundles on projective varieties. Proposed by Jean-Pierre Serre in 1955, the conjecture concerns the relationship between coherent sheaves (or vector bundles) on projective spaces and their behavior when pulled back from smaller-dimensional projective spaces.

In the context of field theory in mathematics, a **splitting field** of a polynomial over a given field is a specific type of field extension that allows the polynomial to factor completely into linear factors.

A `Square` class typically refers to a class used in object-oriented programming to represent a square shape in a geometric context. This class would generally encapsulate properties and behaviors associated with squares, such as their side length, area, perimeter, and possibly methods to manipulate or display the square. Here’s a basic example of what a `Square` class might look like in Python: ```python class Square: def __init__(self, side_length): self.

The Stark conjectures are a set of conjectures in number theory proposed by the mathematician Harold Stark in the 1970s. They are concerned with the behavior of L-functions, particularly the L-functions of certain algebraic number fields, and they provide a profound connection between number theory, the theory of L-functions, and algebraic invariants.

In algebra, "Stufe" typically refers to the term "degree" in English, which indicates the highest power of a variable in a polynomial. The degree of a polynomial is a key concept used to classify polynomials and determine their properties, such as their behavior or the number of roots.

Superreal numbers are an extension of the real numbers which include infinitesimal and infinite quantities. They were introduced in the context of non-standard analysis, a branch of mathematics that studies properties of numbers and functions using hyperreal numbers and other related systems. In more precise terms, superreal numbers can be thought of as a way to incorporate both infinitesimally small and infinitely large quantities into the number system.

The tensor product of fields is a construction that arises in the context of algebra, particularly in the study of vector spaces and modules. Given two fields \( K \) and \( F \), the tensor product \( K \otimes F \) can be viewed in several ways, depending on the context and the mathematical objects you are considering. ### 1. Definition Let \( K \) and \( F \) be two fields.

In the context of mathematics, particularly in algebraic geometry and the study of schemes, the term "thin set" often refers to a certain type of subset of a geometric object that meets specific criteria. However, "Thin set (Serre)" specifically relates to Serre's conjecture (or the Serre's criterion) in the context of schemes.

A totally real number field is a type of number field, which is defined as a finite extension of the field of rational numbers \( \mathbb{Q} \). Specifically, a number field \( K \) is called totally real if every embedding of \( K \) into the complex numbers \( \mathbb{C} \) maps \( K \) into the real numbers \( \mathbb{R} \).

In algebra, a **transcendental extension** refers to a type of field extension that contains elements that are not algebraic over the base field. More formally, if \( K \) is a field, a field extension \( L \) of \( K \) is called a transcendental extension if there exists at least one element \( \alpha \in L \) such that \( \alpha \) is not the root of any non-zero polynomial with coefficients in \( K \).

The Tschirnhaus transformation, named after the German mathematician Ehrenfried Walther von Tschirnhaus, is a mathematical technique used primarily in the field of algebra, particularly in the study of polynomial equations and algebraic curves. This transformation allows one to change the coordinates of a polynomial or algebraic expression to simplify it or transform it into a more convenient form. In particular, the transformation can help eliminate certain terms from a polynomial equation, making it easier to analyze or solve.

Tsen rank, named after mathematician Hsueh-Yung Tsen, is a concept in algebraic geometry and commutative algebra that relates to the behavior of fields and their extensions. Specifically, it provides a measure of the size of a field extension by analyzing the ranks of certain algebraic objects associated with the extension.

In mathematics, particularly in the context of algebra, "U-invariant" typically refers to a property of certain algebraic structures, often in relation to modules or representations over a ring or algebra. In the context of group representation theory, a subspace \( W \) of a vector space \( V \) is said to be U-invariant if it is invariant under the action of the group (or the algebra) associated with \( V \).

A universal quadratic form is a specific type of quadratic form that has the property of representing all possible integers through its integer values. In other words, a quadratic form is called "universal" if it can represent every integer as a value of the form \( ax^2 + bxy + cy^2 \) (for integer coefficients \(a\), \(b\), and \(c\)) for appropriate integer inputs \(x\) and \(y\).

In the context of algebra, "valuation" refers to a function that assigns a value to elements of a certain algebraic structure, often measuring some property of those elements, such as size or divisibility. Valuation is commonly used in number theory and algebraic geometry and can apply to various mathematical objects, such as integers, rational numbers, or polynomials.

A valuation ring is a special type of integral domain that arises in the study of valuation theory in algebraic number theory and algebraic geometry. To understand valuation rings, it's useful to first consider what a valuation is.