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Algebraically constructible real numbers as the smallest real subfield closed under positive square roots
Definition
Let be the intersection of all subfields such that and
This family is nonempty because belongs to it, and the intersection is again a subfield with the same square-root closure. A real number is algebraically constructible when it belongs to .
This is an algebraic definition. It does not assert an equivalence with a physical straightedge-and-compass model.
Depends on
- The rationals form a field
- The reals form a totally ordered field
- Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$
- Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations
Used by
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 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Theorem 1.37 through consequence 1.41 (standard reference, not scraped)