separable polynomial

English

Noun

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 F(ζ) is a normal extension because it is the splitting field of the separable polynomial xn−1 (see Theorem 7.5).
    • 2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:
      Proposition 1.4.2 A finite field extension M/K is Galois if and only if M is the splitting field over K of a separable polynomial.
    • 2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:
      If πP is a separable polynomial in K[t], then the derivative ∂tπP is prime to πP in K[t], and therefore a unit in K[t]P. […] In the case when πP is an inseparable polynomial we may write πP=f(tpr) for a suitable r>0 and separable polynomial f.

Coordinate terms

  • inseparable polynomial

Derived terms

  • discriminantly separable polynomial

Translations

See also

Further reading

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