CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The real cube root of two is algebraic but not algebraically constructible

Statement refuted

Every real algebraic number is algebraically constructible.

Facts & Assumptions

Given: The unique positive real number $a = \sqrt[3]{2}$ .

[L1]

A constructible real algebraic number has degree over $Q$ equal to a power of $2$ (An algebraically constructible real algebraic number has degree over $Q$ equal to a power of two).

[L2]

Eisenstein's criterion proves a primitive integer polynomial irreducible when one prime divides every nonleading coefficient, its square does not divide the constant coefficient, and it does not divide the leading coefficient (Eisenstein criterion over the integers).

[L3]

The degree of a simple algebraic extension equals the degree of the element's minimal polynomial (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L4]

Every nonnegative real has a unique nonnegative $n$ -th root for $n \geq 1$ (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

Counterexample

technique · contradiction

1.1givenL4

By [L4], the real number $a$ exists and satisfies $a^{3} = 2$ , so it is algebraic over $Q$ .

2.1step 1.1L2L3

The polynomial $t^{3} - 2$ is Eisenstein at $2$ , so [L2] makes it irreducible over $Q$ . Hence [L3] gives $[Q (a) : Q] = 3$ .

3.1step 2.1L1assume-contra

Suppose, for contradiction, that $a$ is algebraically constructible. Then [L1] makes its degree a power of $2$ , contrary to step 2.1 because $3$ is odd and greater than $1$ .

4.1step 3.1discharge-contradiction∎

Therefore $a$ is algebraic but not algebraically constructible, refuting the universal statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

35 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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, consequence 1.41 (standard reference, not scraped)