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Concept

Degree of an extension

The dimension of one field regarded as a space of vectors over a smaller one inside it. It multiplies along a tower of extensions, which is what turns a question about constructibility into arithmetic on powers of two.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

computation · Constructible numbers Every rational number that could be a root of x³ − 2. A table of the candidate rational roots allowed by the rational root theorem, with the polynomial's exact value at each.

The cube that will not double

Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.

computation · Constructible numbers Which angles with a rational cosine can be cut in three. A dial of angles marked trisectable or not, beside the cubic whose rational roots decided each one.

The angle that will not divide by three

Halving an angle costs one circle. Cutting it in three means solving a cubic, and for sixty degrees that cubic has no rational root — but plenty of angles do trisect, and which ones is a question with a countable answer.

computation · Constructible numbers Which regular polygons a compass and straightedge can draw, up to 100. A grid of the integers with the constructible ones filled in, each verdict computed two independent ways.

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

computation · Constructible numbers Looking for a polynomial with π as a root. A table of the closest an integer polynomial of each degree comes to vanishing at the number, over a bounded search.

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

computation · Constructible numbers The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

A tower whose degrees multiply

Treat a field containing another as a vector space over it, and the size of an extension becomes a dimension — one that multiplies along a tower, so that three impossible constructions become arithmetic about which numbers divide which.

algebra · Field extensions Seven powers of √2 + ∛3 in a space of six. A table of the powers 1 to (√2 + ∛3)⁶ as coordinate vectors over a six-element basis, with the coefficients of the dependency among them: the minimal polynomial x⁶ − 6x⁴ − 6x³ + 12x² − 36x + 1.

Seven powers in a space of six

Is √2 + ∛3 a root of some polynomial with whole-number coefficients? It lives in a field of dimension six, so its first seven powers are seven vectors in a six-dimensional space and must be dependent — and the dependency, solved exactly, is the polynomial. The same count shows every sum, product and quotient of algebraic numbers is algebraic, without ever needing a formula.

algebra · Field extensions

Named alongside it

The objects these essays reach for when they reach for this one.

Constructible numberMinimal polynomialStraightedge and compassRational root theoremBasisField extensionIrreducible polynomialAlgebraic numberDimensionOperation setClosureConjugate

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