Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Hensel's Lemma Application

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 application of Hensel's lemma in determining nth roots in Ip for all a, focusing on the kernel of Z to Zp. The instructor discusses the image and elements in Z = { x = Zp | 1x1=1} that are nth powers.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (32)

Hybrid kernel

A hybrid kernel is an operating system kernel architecture that attempts to combine aspects and benefits of microkernel and monolithic kernel architectures used in operating systems. The traditional kernel categories are monolithic kernels and microkernels (with nanokernels and exokernels seen as more extreme versions of microkernels). The "hybrid" category is controversial, due to the similarity of hybrid kernels and ordinary monolithic kernels; the term has been dismissed by Linus Torvalds as simple marketing.

Kernel (operating system)

The kernel is a computer program at the core of a computer's operating system and generally has complete control over everything in the system. It is the portion of the operating system code that is always resident in memory and facilitates interactions between hardware and software components. A full kernel controls all hardware resources (e.g. I/O, memory, cryptography) via device drivers, arbitrates conflicts between processes concerning such resources, and optimizes the utilization of common resources e.

Nth root

In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as , where x is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition then of an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: A root of degree 2 is called a square root (usually written without the n as just ) and a root of degree 3, a cube root (written ).

Linux kernel

The Linux kernel is a free and open-source, monolithic, modular, multitasking, Unix-like operating system kernel. It was originally written in 1991 by Linus Torvalds for his i386-based PC, and it was soon adopted as the kernel for the GNU operating system, which was written to be a free (libre) replacement for Unix. Linux is provided under the GNU General Public License version 2 only, but it contains files under other compatible licenses.

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.

Related lectures (32)

Linear Applications: Kernel MOOC: Linear Algebra (Part 2)

Introduces the kernel of a linear application and its properties.

Linear Applications in 3D: Rank Theorem

Explains the Rank Theorem for linear applications in 3D space and its implications.

Quotients de groupe: Un cas particulier

Explores a categorical perspective on group quotients, focusing on a specific case.

The inhomogeneous Ising chain

Covers the inhomogeneous Ising chain, Gibbs measures, and the Borel-Cantelli lemma.

Linear Algebra: Image and Kernel