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

Lecture

Ramified Extensions: Eisenstein Polynomials

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 concept of ramified extensions and their relation to Eisenstein polynomials. Starting with an introduction to OneNote for Windows 10, the instructor explains the process of creating ramified and non-ramified extensions. The lecture delves into the specifics of different types of extensions, such as unramified and topologically ramified ones, and their applications in covering situations and generating covers. By examining examples and roots of Eisenstein polynomials, the lecture illustrates the significance of these extensions in various mathematical contexts.

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)

Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. Branch point In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0.

Branched covering

In mathematics, a branched covering is a map that is almost a covering map, except on a small set. In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle. In algebraic geometry, the term branched covering is used to describe morphisms from an algebraic variety to another one , the two dimensions being the same, and the typical fibre of being of dimension 0.

Eisenstein's criterion

In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort.

Irreducible polynomial

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients.

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.

Related lectures (35)

Local Homeomorphisms and Coverings

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.

Algebraic Curves: Normalization

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Polynomial Factorization and Decomposition

Covers polynomial factorization, irreducible polynomials, ideal decomposition, and the theorem of Bézout.

Generalized Integrals: Elementary Cases

Explores elementary cases of generalized integrals, convergence criteria, and the interpretation of integrals of type i and ii.

Holomorphic Functions: Taylor Series Expansion