CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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The degree of an intermediate field divides the degree of a finite extension

Statement

If $F \subseteq K \subseteq L$ and $L / F$ is finite, then $K / F$ and $L / K$ are finite and

$[K : F] ∣ [L : F] .$

Facts & Assumptions

Given: A tower $F \subseteq K \subseteq L$ with $L / F$ finite.

[L1]

For finite subextensions the tower law is $[L : F] = [L : K] [K : F]$ (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L2]

Assuming the Axiom of Choice, if $S \subseteq V$ spans $V$ then there is a basis $B$ of $V$ with $B \subseteq S$ (Every spanning subset of a vector space contains a basis).

[L3]

A linearly independent subset of a space spanned by $N$ vectors has at most $N$ elements (If $V$ has a spanning set with $n$ elements, then every linearly independent subset of $V$ is finite with at most $n$ elements; in particular $V$ has no linearly independent subset equinumerous with $N$ ).

Proof

technique · direct

1.1

Apply [L2] to the spanning set $K$ of the $F$ -vector space $K$ to obtain an $F$ -basis $B \subseteq K$ . This set is independent in the $F$ -space $L$ , so [L3] makes $B$ finite; hence $K / F$ is finite.

givenL2L3

1.2

Any finite $F$ -basis of $L$ is also a finite $K$ -spanning set of $L$ . Applying [L2] over $K$ gives a finite $K$ -basis, so $L / K$ is finite.

givenL2

2.1

The tower law [L1] now applies and writes $[L : F]$ as $[K : F]$ times the natural number $[L : K]$ , proving the divisibility.

step 1.1step 1.2L1∎

Depends on

Used by

An algebraically constructible real algebraic number has degree over ℚ equal to a power of two Corollary
An algebraic extension need not be finite Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 19 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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)