In algebra, a field k is perfect if any one of the following equivalent conditions holds: Either k has characteristic 0, or, when k has characteristic p > 0, Examples · Perfect closure and perfection

A perfect field is a field F such that every algebraic extension is separable. Any field in field characteristic zero, such as the rationals or the p-adics, ...

Lemma 10.45.2. A field k is perfect if and only if it is a field of characteristic 0 or a field of characteristic p > 0 such that every element has a pth root.

A field K is called perfect if every irreducible polynomial in K[T] is separable. Every field of characteristic 0 is perfect. We will see that finite fields are ...

21 июл. 2017 г. · Definition. A field (in the sense of commutative algebra) F F is perfect if every algebraic extension of F F is separable.

2 февр. 2024 г. · I am taking undergrad level field theory and I am doing a problem: Suppose K[x] is a polynomial ring over the field K and F is a subfield of K. abstract algebra - Examples of fields which are not perfect Every algebraic extension of a perfect field is separable and perfect About perfect fields - abstract algebra - Mathematics Stack Exchange Perfect field of charateristic p, equivalence of definitions of a perfect ... Другие результаты с сайта math.stackexchange.com

6 нояб. 2023 г. · We define perfect fields using the second characterization and show the equivalence to the first characterization. The implication ``2 => 1'' is ...

11 окт. 2014 г. · A field k over which every polynomial is separable. In other words, every algebraic extension of k is a separable extension.

22 мар. 2013 г. · All fields of characteristic 0 are perfect, so in particular the fields R ℝ , C ℂ and Q ℚ are perfect. If K K is a field of characteristic p ...