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

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

     Proposition 1.4.2 A finite field extension $M/K$ 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 $\pi _{P}$ is a separable polynomial in $K[t]$, then the derivative $\partial _{t}\pi _{P}$ is prime to $\pi _{P}$ in $K[t]$, and therefore a unit in $K[t]_{P}$. […] In the case when $\pi _{P}$ is an inseparable polynomial we may write $\pi _{P}=f(t^{p^{r}})$ for a suitable $r>0$ and separable polynomial $f$.

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) on Wikipedia.Wikipedia
  • Formal derivative on Wikipedia.Wikipedia
• Separable Polynomial on Wolfram MathWorld
• Separable extension on Encyclopedia of Mathematics