Separable polynomial

Let $𝐾$ be a field. A polynomial $𝑓 (𝑥) \in 𝐾 [𝑥]$ is separable iff all its roots have multiplicity 1 in $𝐹 [𝑥]$ , where $𝐹$ is its splitting field (or algebraic closure). #m/def/field Equivalently,

g c d 𝐾 [𝑥] {𝑓 (𝑥), 𝑑 𝑑 𝑥 𝑓 (𝑥)} = 1

Otherwise $𝑓 (𝑥)$ is called inseparable.¹

Proof of equivalence

Suppose $𝑓 (𝑥) \in 𝐾 [𝑥]$ is inseparable, i.e. it has a multiple root in a splitting field $𝐹$ , i.e.

𝑓 (𝑥) = (𝑥 - 𝛼) 𝑚 𝑔 (𝑥)

for some $𝛼 \in 𝐹$ , $𝑔 (𝑥) \in 𝐹 [𝑥]$ , $𝑚 \geq 2$ . It follows

𝑓' (𝑥) = 𝑚 (𝑥 - 𝛼) 𝑚 - 1 𝑔 (𝑥) - (𝑥 - 𝛼) 𝑚 𝑔' (𝑥),

so $𝛼$ is a common root of $𝑓 (𝑥)$ and $𝑓' (𝑥)$ . Thus both are divisible by the minimal polynomial $ℎ (𝑥) \in 𝐾 [𝑥]$ of $𝛼$ , so

g c d 𝐾 [𝑥] {𝑓 (𝑥), 𝑑 𝑑 𝑥 𝑓 (𝑥)} \neq 1 .

For the converse, suppose

g c d 𝐾 [𝑥] {𝑓 (𝑥), 𝑑 𝑑 𝑥 𝑓 (𝑥)} \neq 1 .

so that in particular, $𝑓 (𝑥)$ and $𝑓' (𝑥)$ have a common root $𝛼 \in ―― 𝐾$ in the algebraic closure. Write $𝑓 (𝑥) = (𝑥 - 𝛼) ℎ (𝑥)$ where $ℎ (𝑥) \in ―― 𝐾 (𝑥)$ and thus

𝑓' (𝑥) = ℎ (𝑥) + (𝑥 - 𝛼) ℎ' (𝑥)

whence $(𝑥 - 𝛼) ∣ ℎ (𝑥)$ and thus $(𝑥 - 𝛼) 2 ∣ 𝑓 (𝑥)$ , so $𝑓 (𝑥)$ is inseparable.

Properties

If $𝑓 (𝑥) \in 𝐾 [𝑥]$ is an inseparable irreducible polynomial, then $𝑓' (𝑥) = 0$ .
If $c h a r 𝐾 = 0$ , all irreducible polynomials are inseparable.

Proof of 1

Since $𝑓 (𝑥)$ is inseparable, $𝑓 (𝑥)$ and $𝑓' (𝑥)$ have a common irrreducible factor $𝑞 (𝑥)$ , but since $𝑓 (𝑥)$ is irreducible, $𝑞 (𝑥)$ and $𝑓 (𝑥)$ must be associate and thus of the same degree. But since $𝑞 (𝑥) ∣ 𝑓' (𝑥)$ , it follows $𝑓' (𝑥) = 0$ , proving ^P1. ^P2 is an immediate corollary.

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

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

Separable polynomial

Properties

Footnotes