separable polynomial

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Noun

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separable polynomial (plural separable polynomials)

  1. (algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial).
    Over a perfect field, the separable polynomials are precisely the square-free polynomials.
    The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.
    • 1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:
      We know that is a normal extension because it is the splitting field of the separable polynomial (see Theorem 7.5).
    • 2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:
      Proposition 1.4.2 A finite field extension is Galois if and only if is the splitting field over of a separable polynomial.
    • 2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:
      If is a separable polynomial in , then the derivative is prime to in , and therefore a unit in . [] In the case when is an inseparable polynomial we may write for a suitable and separable polynomial .

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