False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-16

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FALSE: degrees add in a tower of finite field extensions

Statement

For every finite tower $F \subseteq K \subseteq L$ , one has

$[L : F] = [L : K] + [K : F] .$

Facts & Assumptions

Given: A field $L$ of order $64 = 2^{6}$ .

[L1]

The tower law is multiplicative: $[L : F] = [L : K] [K : F]$ (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L2]

The field $F_{64}$ has a unique subfield $K$ of order $4$ and prime subfield $F$ of order $2$ (The subfields of $F_{p^{n}}$ are the unique fields $F_{p^{d}}$ for positive divisors $d$ of $n$ ).

[L3]

A field of order $64$ exists (For every prime $p$ and $n \geq 1$ , a field with $p^{n}$ elements exists).

Refutation

technique · direct

1.1givenL2L3

Choose $L = F_{64}$ using [L3], let $F = F_{2}$ , and let $K = F_{4} \subseteq L$ from [L2]. Then $[K : F] = 2$ and $[L : F] = 6$ .

2.1step 1.1L1algebra

By the true tower law [L1], $6 = [L : K] \cdot 2$ , so $[L : K] = 3$ .

3.1step 1.1step 2.1∎

The false additive formula would give $[L : F] = 3 + 2 = 5$ , contradicting the actual value $6$ . Thus degrees multiply, rather than add, in a tower.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Finite Fields, Sections 1-2 (standard reference, not scraped)