Field extension
Named by 17 essays across 3 fields — each of them below, with the objects they name alongside it.
What two points can build
A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.
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.
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.
The lattice that runs the other way
The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.
The mark that changes what is reachable
Two thousand years of failure to trisect an angle with compass and straightedge was failure at a stated set of operations. Scratch two marks on the straightedge and Archimedes trisects any angle in four steps — because the new operation solves a cubic, and the old ones could only ever solve quadratics.
Two instruments with one reach
Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.
A quintic a sliding mark reaches
The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.
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.
![The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.](/f/the-integers-of-q-5-with-z-5-inside-them-a4f067e7.svg)
The integers a field contains
Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.
How a polynomial breaks modulo the primes
Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.
Units that form a lattice
In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.
The lengths dividers cannot reach
A pair of dividers carries a length from one place to another and draws nothing. With a straightedge it finds midpoints, parallels and right angles, and it draws the regular 17-gon. It cannot draw a segment of length √(1 + √2) — and the reason is not on the page at all, but in the other root of the equation that number solves.
The sums that obey a smaller equation
The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.
Three real roots and no real radicals
The cubic x³ − 3x + 1 has three real roots, and Cardano's formula reaches every one of them by way of the cube roots of a complex number. That detour cannot be removed: Hölder proved in 1891 that no expression built from real radicals gives a root of an irreducible cubic whose roots are all real. The proof is three lines of the Galois correspondence, and its general form says that real radicals reach all-real roots only when square roots alone would.
The five crescents that can be squared
The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.
What two points build in two rounds
Start with two points and draw every line and circle they allow; mark every crossing; do it again. The first round gives six points, the second gives 203, and the third gives over 1.7 billion. Computed exactly, the 203 points show the tower of square roots growing in real time — eleven rational points, seventy-six that need √3, and the rest needing one of six new square roots.
Two fields that every prime confuses
Reduce x⁸ − 3 and x⁸ − 48 modulo any prime other than 2 and 3 and factor them: the two lists of factor degrees are the same, at all 9,590 such primes below 100,000. The fields the two equations define are still different. The reason is a pair of ways for one group to permute eight roots, with the same cycle lengths element by element and different fixed points — the same coincidence, in degree seven, as the points and lines of the Fano plane.
Named alongside it
The objects these essays reach for when they reach for this one.
Constructible numberGalois groupDegreeOperation setIrreducible polynomialMinimal polynomialBasisClosureConjugateCubicDegree of an extensionLattice