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Separability is transitive in towers of algebraic extensions

Statement

Let $F \subseteq K \subseteq L$ be algebraic field extensions. If $K / F$ and $L / K$ are separable, then $L / F$ is separable.

Facts & Assumptions

Given: An algebraic tower $F \subseteq K \subseteq L$ with $K / F$ and $L / K$ separable, and an element $a \in L$ .

[L1]

A finite extension is separable exactly when its separable degree equals its ordinary degree (A finite extension is separable if and only if $[K : F]_{s} = [K : F]$ ).

[L2]

Separable degree is multiplicative in finite towers (Separable degree is multiplicative in finite towers: $[L : F]_{s} = [L : K]_{s} [K : F]_{s}$ ).

[L3]

A simple extension has full separable degree exactly when its minimal polynomial has all roots distinct (The separable degree of $F (α) / F$ is the number of distinct roots of $m_{α}$ ).

[L4]

Polynomial gcd is unchanged after extending the coefficient field (The monic gcd of two base-field polynomials is unchanged after extending the coefficient field).

[L5]

Ordinary degrees multiply in finite towers (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L6]

Finitely many algebraic generators produce a finite extension (An extension generated by finitely many algebraic elements is finite).

[L7]

Separability is the elementwise separability of minimal polynomials (Separable algebraic elements and separable extensions).

Proof

technique · direct

1.1

Let $f \in K [x]$ be the minimal polynomial of $a$ over $K$ , and let $E = F (c_{0}, \dots, c_{r}) \subseteq K$ be generated by its coefficients. The $c_{i}$ are separable over $F$ by hypothesis and $E / F$ is finite by [L6]. Adjoining the $c_{i}$ successively, each relative minimal polynomial divides a separable minimal polynomial over $F$ , so [L3], [L2], and [L5] give $[E : F]_{s} = [E : F]$ .

L2L3L5L6L7

1.2

The polynomial $f$ is separable over $K$ because $a$ is separable over $K$ . By gcd stability [L4], it is already coprime to its derivative in $E [x]$ ; hence every irreducible factor over $E$ , in particular the minimal polynomial of $a$ over $E$ , is separable. Thus $E (a) / E$ has full separable degree by [L3].

L3L4L7

2.1

Multiplicativity [L2] and the ordinary tower law [L5] now give $[E (a) : F]_{s} = [E (a) : F]$ . By [L1], $E (a) / F$ is separable, so its element $a$ is separable over $F$ .

step 1.1step 1.2L1L2L5

3.1

Since $a \in L$ was arbitrary, [L7] makes $L / F$ separable. Trivial steps of the tower are included because their degree and separable degree are both one.

step 2.1L7∎

Depends on

Used by

Assuming Choice, separable closures exist and are base-isomorphic Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. L. Clark, Field Theory, Chapters 4 and 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)