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Lecture

Norm Extension in Finite Fields

In course

MATH-494: Topics in arithmetic geometry

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Ze ...

Description

This lecture covers the uniqueness of norm extension in finite fields, focusing on the Galois property and prime fields. It explains the construction of norms on finite extensions of Qp, emphasizing the existence of at most one field norm extending the one on F. The instructor discusses the concept of norms on finite extensions, the cyclic group of order q-1, and the extension of norms. The lecture concludes with the definition of the norm of a field-linear endomorphism and the application of norms in finite fields.

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Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra, Polynomials

Related concepts (32)

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power.

Windows 10

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Windows 11

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