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). 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, then .
2. If [[Characteristic|]], 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.

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Footnotes

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