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.
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 \( R \).
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.
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.
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.
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 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.
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.
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.
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 generic polynomial is a polynomial that is defined with coefficients that can represent any number, typically treated as indeterminate or symbolic variables.
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.
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.
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 \( K \).
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.
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.
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.
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.
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.
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.