Concept

Transcendence

The property of a number that no polynomial with whole-number coefficients has it as a root. Establishing it for a particular number is hard, and it is what puts a length beyond the reach of straightedge and compasses.

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

$8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.$

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

number · pigeonhole

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

Named alongside it

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

Basis Constructible number Continued fraction convergent Continued fractions Counting argument Degree of an extension Dimension Dirichlet Existence proof Field extension Golden ratio Irreducible polynomial

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