separable polynomial

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 ${\\displaystyle F(\\zeta )}$ 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 ${\\displaystyle K[t]}$, then the derivative ${\\displaystyle \\partial _{t}\\pi _{P}}$ is prime to ${\\displaystyle \\pi _{P}}$ in ${\\displaystyle K[t]}$, and therefore a unit in ${\\displaystyle K[t]_{P}}$. […] 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 ${\\displaystyle f}$.

Coordinate terms

• inseparable polynomial

Derived terms

• discriminantly separable polynomial

Related terms

• separable extension

Translations

polynomial that has distinct roots

Italian: polinomio separabile m Spanish: polinomio separable m

See also

• splitting field
• square-free polynomial

Further reading

• Separable extension on Wikipedia.Wikipedia
• Perfect field on Wikipedia.Wikipedia
• Square-free polynomial on Wikipedia.Wikipedia
• Multiplicity (mathematics)#Multiplicity of a root of a polynomial on Wikipedia.Wikipedia
  • Formal derivative#Application to finding repeated factors on Wikipedia.Wikipedia
• Separable Polynomial on Wolfram MathWorld
• Separable extension on Encyclopedia of Mathematics