Lecture

Algebraic Fields: Transcendence Degree

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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 concept of transcendence degree in algebraic fields, focusing on the cardinality and properties of sets like Qp, Qp, and Cp. The instructor explains the cardinality of sets, the continuity of roots of polynomials, and the isomorphism of fields with the same characteristic. The lecture delves into the behavior of classical analytic functions, such as logarithmic and exponential functions, and demonstrates proofs related to irrationality using algebraic manipulations. The presentation concludes with a discussion on transcendence degrees and the limitations of certain mathematical proofs.

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Related concepts (32)

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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.

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