separable polynomial

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separable polynomial

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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 ${\displaystyle F(\zeta )}$ is a normal extension because it is the splitting field of the separable polynomial ${\displaystyle x^{n}-1}$ (see Theorem 7.5).

   • 2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:

     Proposition 1.4.2 A finite field extension ${\displaystyle M/K}$ is Galois if and only if ${\displaystyle M}$ is the splitting field over ${\displaystyle K}$ of a separable polynomial.

   • 2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:

     If ${\displaystyle \pi _{P}}$ is a separable polynomial in ${\displaystyle K}$, then the derivative ${\displaystyle \partial _{t}\pi _{P}}$ is prime to ${\displaystyle \pi _{P}}$ in ${\displaystyle K}$, and therefore a unit in ${\displaystyle K_{P}}$. […] In the case when ${\displaystyle \pi _{P}}$ is an inseparable polynomial we may write ${\displaystyle \pi _{P}=f(t^{p^{r}})}$ for a suitable ${\displaystyle r>0}$ and separable polynomial ${\displaystyle f}$.

• inseparable polynomial

• discriminantly separable polynomial

• separable extension

• splitting field
• square-free polynomial

• Separable extension on Wikipedia.Wikipedia
• Perfect field on Wikipedia.Wikipedia
• Square-free polynomial on Wikipedia.Wikipedia
• Multiplicity (mathematics) on Wikipedia.Wikipedia
  • Formal derivative on Wikipedia.Wikipedia
• Separable Polynomial on Wolfram MathWorld
• Separable extension on Encyclopedia of Mathematics