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Concept

Perfect field

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 ↦ x''p'' 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., A \otimes_k F 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 theor

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Summary

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 ↦ x''p'' 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., A \otimes_k F 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 theor

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Related concepts (12)

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Separable extension

In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the op

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped wit

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic p > 5. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic p > 5 are rational. We show that these theorems are sharp by providing counterexamples in characteristic 5.

CAMBRIDGE UNIV PRESS2022

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

Emelie Kerstin Arvidsson

We investigate the vanishing of H1(X,OX(−D)) for a big and nef Q-Cartier Z-divisor D on a log del Pezzo pair (X,Δ) over a perfect field of positive characteristic p.

2021

Vanishing theorems in positive characteristic

Emelie Kerstin Arvidsson

The topic of this thesis is vanishing theorems in positive characteristic. In particular, we use "the covering trick of Ekedahl" to investigate the vanishing of

H^1(X, \mathcal{O}_X(-D))

for a big and nef Weil divisor

D

on a normal projective variety with

-K_X

nef. In dimension two, we show that on a surface of log del Pezzo type over a perfect field of characteristic

p>5

this vanishing holds. More generally, using techniques of the \emph{Minimal model program} we prove the Kawamata--Viehweg vanishing theorem in this setting. We also construct a counter-example in characteristic five, showing that our result is optimal. We discuss the relationship (due to Hacon--Witaszek) between this vanishing theorem and properties of threefold klt-singularities. We investigate if a similar relationship exists between threefold lc-singularities and a certain vanishing theorem for higher direct images of elliptic fibrations. This leads to a counter-example to a theorem of Koll'ar over the complex numbers, in every positive characteristic.

EPFL2021