Minimal polynomial
Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
On the circle and never home
Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.
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.
The sums of roots of unity that add to nothing
All n of the n-th roots of unity add to zero, and so does any regular polygon among them, turned. Those are not the only vanishing sums: six thirtieths of a turn close into a loop with no polygon in them. Which counts of roots can close at all is decided by the prime factors of n — no seven fifteenths ever add to zero — and the same question counts where the diagonals of a regular polygon cross.
The class number is a degree
The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.
How close two sums of square roots come
Two different sums of square roots of whole numbers are never equal: square roots of different squarefree numbers are independent, so no cancellation can make them agree. But they can come very close. Among sums of two square roots of numbers up to 320, √88 + √289 and √171 + √177 agree to eight decimal places. How many digits must be computed, in the worst case, to tell two such sums apart is not known — the best proof allows a number of digits doubly exponential in the length of the sums, the examples suggest only a modest polynomial, and the gap between them is why nobody knows whether the shortest route through points in the plane can be checked quickly.
Named alongside it
The objects these essays reach for when they reach for this one.
Degree of an extensionConstructible numberField extensionAlgebraic integerAlgebraic numberBasisConjugateStraightedge and compassCyclotomic polynomialDimensionIrreducible polynomialRational root theorem