Corpo de Levi-Civita

Corpo de Levi-Civita, em matemática, é um corpo descrito [Nota 1] pelo jovem matemático Tullio Levi-Civita, como um corpo ordenado que contém elementos infinitesimais e é Cauchy completo.[1]

Um elemento x deste corpo pode ser escrito como a série formal de potências:

em que qj são números racionais crescentes e xq são números reais.[2][Nota 2]

Neste corpo pode ser definida uma relação de ordem, e, para elementos positivos, é possível definir quando um número é infinitamente maior (ou menor) que outro: a > 0 é infinitamente menor que b > 0 (escreve-se a << b) quando, qualquer que seja n natural, n . a < b. Existem elementos infinitesimais e elementos infinitamente grandes neste corpo.[3][2]

Neste corpo, com a topologia induzida pela ordem, toda sequência de Cauchy converge.[4][2]

Neste corpo, assim como no corpo dos números reais, todo número positivo tem duas raízes quadradas, nenhum número negativo tem raiz quadrada, e todo número tem uma única raiz n-ésima, para n ímpar.[5] O corpo é um corpo real fechado,[2] ou seja, todo polinômio de grau ímpar tem raiz e todo número positivo tem raiz quadrada.[Nota 3]

Este corpo é a menor [Nota 4] extensão dos reais que é um corpo ordenado não arquimediano, Cauchy completo e real fechado.[2]

Notas e referências

Notas

  1. ↑ O texto de Martin Berz diz que o corpo foi descoberto por Levi-Civita; a ideia de que objetos matemáticos existem e são descobertos é chamada de platonismo.
  2. ↑ O texto de Shamseddine apresenta esta série como um somatório, e usa d em vez de ε; aqui foi usado ε pois este símbolo é, intuitivamente, associado a um infinitesimal.
  3. ↑ Conforme artigo corpo real fechado.
  4. ↑ A menos de isomorfismo.

Referências

  1. ↑ Martin Berz, 2. Calculus and Numerics on Levi-Civita Fields, 1. Introduction, p.21 [em linha]
  2. ↑ a b c d e Khodr Shamseddine, Advances in p-adic and Non-archimedian Analysis (2010), p.219 [google books visualização parcial]
  3. ↑ Martin Berz, 2. Calculus and Numerics on Levi-Civita Fields, 3. Order Structure, p.27
  4. ↑ Martin Berz, 2. Calculus and Numerics on Levi-Civita Fields, 4. Topology, Convergence, and Cauchy-Completeness, p.28
  5. ↑ Martin Berz, 2. Calculus and Numerics on Levi-Civita Fields, 2. Algebraic Properties of R, p.26


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