Lecture

Galois Theory of Qp

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

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 Galois theory of Qp, focusing on understanding algebraic extensions of Qp. The instructor explains various facts about the Galois theory at Qp, including the concept of inertia groups and cyclic properties. The lecture delves into the relationship between Galois extensions and residue fields, highlighting the complexity and uniqueness of certain groups. Additionally, the discussion extends to local and global fields, emphasizing the significance of complete fields in the context of Galois theory. The session concludes with insights into Krasner's Lemma and its applications, showcasing the independence of arguments from specific fields like Qp.

Official source

Ontological neighbourhood

Mathematics

Algebra: Polynomials

Related concepts (32)

Windows 10

Windows 10 is a major release of Microsoft's Windows NT operating system. It is the direct successor to Windows 8.1, which was released nearly two years earlier. It was released to manufacturing on July 15, 2015, and later to retail on July 29, 2015. Windows 10 was made available for download via MSDN and TechNet, as a free upgrade for retail copies of Windows 8 and Windows 8.1 users via the Windows Store, and to Windows 7 users via Windows Update.

Windows 10 editions

Windows 10 has several editions, all with varying feature sets, use cases, or intended devices. Certain editions are distributed only on devices directly from an original equipment manufacturer (OEM), while editions such as Enterprise and Education are only available through volume licensing channels. Microsoft also makes editions of Windows 10 available to device manufacturers for use on specific classes of devices, including IoT devices and previously marketed Windows 10 Mobile for smartphones.

Start menu

The Start menu is a graphical user interface element that has been part of Microsoft Windows since Windows 95, providing a means of opening programs and performing other functions in the Windows shell. The Start menu, and the Taskbar on which it appears, were created and named in 1993 by Daniel Oran, a program manager at Microsoft who had previously collaborated on Great ape language research with the behavioral psychologist B.F. Skinner at Harvard. The Start menu was renamed Start screen in Windows 8, before returning to its original name with Windows 10.

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

Microsoft Office

Microsoft Office, or simply Office, is a discontinued family of client software, server software, and services developed by Microsoft. It was first announced by Bill Gates on August 1, 1988, at COMDEX in Las Vegas. Initially a marketing term for an office suite (bundled set of productivity applications), the first version of Office contained Microsoft Word, Microsoft Excel, and Microsoft PowerPoint. Over the years, Office applications have grown substantially closer with shared features such as a common spell checker, Object Linking and Embedding data integration and Visual Basic for Applications scripting language.

Related lectures (59)

Hensel's Lemma and Field Theory

Covers the proof of Hensel's Lemma and a review of field theory, including Newton's approximation and p-adic complex numbers.

Norm Extension in Finite Fields

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

Galois Theory: The Galois Correspondence

Explores the Galois correspondence and solvability by radicals in polynomial equations.

Galois Theory: Dedekind Rings

Explores Galois theory with a focus on Dedekind rings and their unique factorization of fractional ideals.

Ramification and Structure of Finite Extensions