Ring theory | Field (mathematics)
In algebra, a field k is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over k has distinct roots. * Every irreducible polynomial over k is separable. * Every finite extension of k is separable. * Every algebraic extension of k is separable. * Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. * Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ xp is an automorphism of k. * The separable closure of k is algebraically closed. * Every reduced commutative k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below) Otherwise, k is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above). Another important property of perfect fields is that they admit Witt vectors. More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism. (When restricted to integral domains, this is equivalent to the above condition "every element of k is a pth power".) (Wikipedia).
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From playlist How to videos!
From playlist the absolute best of stereolab
What Is a Field? - Instant Egghead #42
Contributing editor George Musser explains how physicists think about the universe using the fundamental concept of "the field". -- WATCH more Instant Egghead: http://goo.gl/CkXwKj SUBSCRIBE to our channel: http://goo.gl/fmoXZ VISIT ScientificAmerican.com for the latest science news:http
From playlist Quantum Field Theory
An introduction to perfectoid spaces and the tilting... - M. Morrow - Workshop 2 - CEB T1 2018
Matthew Morrow (CNRS – Sorbonne Université) / 09.03.2018 An introduction to perfectoid spaces and the tilting correspondence. This expository survey will aim to provide an introduction to Scholze’s formalism of tilting, which serves as a sort of transfer principle through which p-adic pr
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Perfectoid spaces (Lecture 3) by Kiran Kedlaya
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Perfectoid spaces (Lecture 1) by Kiran Kedlaya
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Pushing back the barrier of imperfection - F-V. Kuhlmann - Workshop 2 - CEB T1 2018
Franz-Viktor Kuhlmann (Szczecin) / 06.03.2018 The word “imperfection” in our title not only refers to fields that are not perfect, but also to the defect of valued field extensions. The latter is not necessarily directly connected with imperfect fields but may always appear when at least
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
NIP Henselian fields - F. Jahnke - Workshop 2 - CEB T1 2018
Franziska Jahnke (Münster) / 05.03.2018 NIP henselian fields We investigate the question which henselian valued fields are NIP. In equicharacteristic 0, this is well understood due to the work of Delon: an henselian valued field of equicharacteristic 0 is NIP (as a valued field) if and on
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
From playlist the absolute best of stereolab
Perfectoid spaces (Lecture 2) by Kiran Kedlaya
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Jacob Lurie: A Riemann-Hilbert Correspondence in p-adic Geometry Part 2
At the start of the 20th century, David Hilbert asked which representations can arise by studying the monodromy of Fuchsian equations. This question was the starting point for a beautiful circle of ideas relating the topology of a complex algebraic variety X to the study of algebraic diffe
From playlist Felix Klein Lectures 2022
WordPress Plugin Development - Part 33 - Create a Custom Taxonomy Manager
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From playlist WordPress Plugins Development Tutorials
Peter SCHOLZE (oct 2011) - 3/6 Perfectoid Spaces and the Weight-Monodromy Conjecture
We will introduce the notion of perfectoid spaces. The theory can be seen as a kind of rigid geometry of infinite type, and the most important feature is that the theories over (deeply ramified extensions of) Q_p and over F_p((t)) are equivalent, generalizing to the relative situation a th
From playlist Peter SCHOLZE (oct 2011) - Perfectoid Spaces and the Weight-Monodromy Conjecture
Bonus track that can be found on some limited edition issues of Chemical Chords, especially Japanese ones.
From playlist the absolute best of stereolab