Field theory MOC

Separable polynomial

Let be a field. A polynomial is separable iff all its roots have multiplicity 1 in , where is its splitting field (or algebraic closure). #m/def/field Equivalently,

Otherwise is called inseparable.1

Proof of equivalence

Suppose is inseparable, i.e. it has a multiple root in a splitting field , i.e.

for some , , . It follows

so is a common root of and . Thus both are divisible by the minimal polynomial of , so

For the converse, suppose

so that in particular, and have a common root in the algebraic closure. Write where and thus

whence and thus , so is inseparable.

See also Separable extension.

Properties

  1. If is an inseparable irreducible polynomial, then .
  2. If , all irreducible polynomials are inseparable.
Proof of 1

Since is inseparable, and have a common irrreducible factor , but since is irreducible, and must be associate and thus of the same degree. But since , it follows , proving ^P1. ^P2 is an immediate corollary.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2009. Algebra: Chapter 0, §VII.4.2, p. 434 ↩