separable polynomial

plural separable polynomials

1. field theoryalgebraA 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.

   4 more examples

   The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.

   We know that F(#92;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).

   1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277

   Proposition 1.4.2 A finite field extension M#47;K is Galois if and only if M is the splitting field over K of a separable polynomial.

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

   If #92;pi#95;P is a separable polynomial in K#91;t#93;, then the derivative #92;partial#95;t#92;pi#95;P is prime to #92;pi#95;P in K#91;t#93;, and therefore a unit in K#91;t#93;#95;P.[…]In the case when #92;pi#95;P is an inseparable polynomial we may write #92;pi#95;P#61;f(t#123;pʳ#125;) for a suitable rgt;0 and separable polynomial f.

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