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An algebraically constructible real algebraic number has degree over $Q$ equal to a power of two

Statement

If a real algebraic number $x$ is algebraically constructible, then

$[Q (x) : Q] = 2^{s}$

for some $s \in N$ .

Facts & Assumptions

Given: A real algebraic, algebraically constructible number $x$ .

[L1]

The element $x$ lies in a finite tower of quadratic extensions beginning at $Q$ (A real number is algebraically constructible exactly when it lies in a finite tower of real quadratic adjunctions).

[L2]

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

[L3]

The degree of an intermediate field divides the total finite degree (The degree of an intermediate field divides the degree of a finite extension).

[L4]

Prime factorization is unique, so a positive divisor of a power of $2$ is itself a power of $2$ (For $n \geq 1$ and any injective list $p : r \to Z$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i < r} p_{i}^{v_{p_{i}} (n)}$ ; the exponents are determined by $n$ , and $v_{q} (n) = 0$ for every prime $q$ outside the list).

Proof

technique · direct

1.1

Choose from [L1] a tower $Q = K_{0} \subseteq \dots \subseteq K_{r}$ with $x \in K_{r}$ and every step degree $2$ .

givenL1choose

2.1

Repeated application of [L2] gives $[K_{r} : Q] = 2^{r}$ .

step 1.1L2algebra

3.1

The simple field $Q (x)$ is intermediate between $Q$ and $K_{r}$ , so [L3] says its degree divides $2^{r}$ .

step 1.1step 2.1L3

4.1

By [L4], that positive divisor is $2^{s}$ for some $s \leq r$ .

step 3.1L4∎

Depends on

Used by

The real cube root of two is algebraic but not algebraically constructible Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 115 results over 25 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 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, consequence 1.41 (standard reference, not scraped)

An algebraically constructible real algebraic number has degree over QQ equal to a power of two