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Field extensions — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · algebra
  2. 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.

    part 2 · algebra
  3. 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.

    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.

    part 3 · algebra
  4. How often each factor pattern occurs, against the Galois group. Bars for four polynomials — x³ − 3x + 1, x³ − 2, x⁴ − 10x² + 1, x⁴ − 2 — giving the share of primes up to 20000 with each factorisation pattern, beside the predicted share from each Galois group.

    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.

    part 4 · algebra
  5. The units of a cubic field, as a lattice of logarithms. Points for 66 units of the field of a root of x³ − 3x + 1, plotted by the logarithms of two of their conjugates, lying on the lattice spanned by the logarithms of θ and θ − 1.

    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.

    part 5 · algebra

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