Field norm | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite dimensional vector space over K. Multiplication by α, an element of L, is a K-linear transformation of this vector space into itself. The norm, NL/K(α), is defined as the determinant of this linear transformation. If L/K is a Galois extension, one may compute the norm of α ∈ L as the product of all the Galois conjugates of α: where Gal(L/K) denotes the Galois group of L/K. (Note that there may be a repetition in the terms of the product.) For a general field extension L/K, and nonzero α in L, let σ_1(α), ..., σ_n(α) be the roots of the minimal polynomial of α over K (roots listed with multiplicity and lying in some extension field of L); then If L/K is separable, then each root appears only once in the product (though the exponent, the degree [L:K(α)], may still be greater than 1). One of the basic examples of norms comes from quadratic field extensions where is a square-free integer. Then, the multiplication map by on an element is The element can be represented by the vector since there is a direct sum decomposition as a -vector space. The matrix of is then and the norm is , since it is the determinant of this matrix. Consider the number field . The Galois group of over has order and is generated by the element which sends to . So the norm of is: The field norm can also be obtained without the Galois group. Fix a -basis of , say: Then multiplication by the number sends 1 to and to . So the determinant of "multiplying by " is the determinant of the matrix which sends the vector (corresponding to the first basis element, i.e., 1) to , (corresponding to the second basis element, i.e., ) to , viz.: The determinant of this matrix is −1. Another easy class of examples comes from field extensions of the form where the prime factorization of contains no -th powers, for a fixed odd prime.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (6)

Related concepts (10)

Related lectures (5)

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers. Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.

Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry.

Computer Algebraic Approach to Verification and Debugging of Galois Field Multipliers

Cunxi Yu, Maciej Jerzy Ciesielski

The paper presents a novel method to verify and debug gate-level arithmetic circuits implemented in Galois Field arithmetic. The method is based on forward reduction of the specification polynomials of the circuit in GF(2(m)) using GF(2) models of its logi ...

IEEE2018

Small subset sums

Gergely Ambrus

Let parallel to.parallel to be a norm in R-d whose unit ball is B. Assume that V subset of B is a finite set of cardinality n, with Sigma(v is an element of V) v = 0. We show that for every integer k with 0

Elsevier Science Inc2016

Algebraic algorithms for network information flow

Javad Ebrahimi Boroojeni

Ahlswede et al. in the seminal paper [1] have shown that in data transfer over networks, processing the data at the nodes can significantly improve the throughput. As proved by Li et al. in [2], even with a simple type of operation, namely linear operation ...

EPFL2013

Explores Frobenius theorems in number theory, ideal class groups, norm properties, and geometry of numbers.

Covers the extension of fields, defining the norm of an element from one field to another, and introducing the p-adic valuation.

Covers the uniqueness of norm extension in finite fields and the construction of norms on finite extensions of Qp.