Meaning of separable polynomial | Babel Free

Definitions

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).

Equivalents

Examples

“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.”

“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).”

“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.”

“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.”

CEFR level