LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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A simple finite extension has only finitely many intermediate fields

Statement

If $E / F$ is a finite simple extension, then there are only finitely many intermediate fields $F \subseteq M \subseteq E$ .

Facts & Assumptions

Given: A finite simple extension $E = F (α)$ .

[L1]

The notation $F (α)$ denotes the smallest subfield containing $F$ and $α$ (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

[L2]

An algebraic element has a unique monic irreducible minimal polynomial over its base field (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

[L3]

A polynomial ring over a field is a unique factorisation domain (For every field $F$ , $F [x]$ is a unique factorisation domain).

[L4]

Degrees multiply in a finite tower of field extensions (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

Proof

technique · direct

1.1

Let $f$ be the minimal polynomial of $α$ over $F$ . For an intermediate field $M$ , let $g_{M}$ be the minimal polynomial of $α$ over $M$ and let $M_{0}$ be the subfield of $M$ generated over $F$ by the coefficients of $g_{M}$ .

L1L2

2.1

The polynomial $g_{M}$ divides $f$ in $M [x]$ and therefore in $E [x]$ . It is irreducible over $M_{0}$ , since a factorisation over $M_{0}$ would be one over $M$ , so it is also the minimal polynomial of $α$ over $M_{0}$ .

step 1.1L2

3.1

Thus $[E : M_{0}] = \deg g_{M} = [E : M]$ ; the tower law [L4] in $M_{0} \subseteq M \subseteq E$ gives $[M : M_{0}] = 1$ , so $M = M_{0}$ . Hence the coefficients of $g_{M}$ determine $M$ .

step 2.1L4

4.1

By unique factorisation [L3], the fixed polynomial $f$ has only finitely many monic divisors in $E [x]$ . The injective assignment $M \mapsto g_{M}$ therefore proves that there are only finitely many intermediate fields.

step 3.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 46 results over 11 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 3 to 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapter 5 (standard reference, not scraped)