Algebra — page 3

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.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

One number under every bell

The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.

Every power sum, from the coefficients alone

Raise the roots of a polynomial to the k-th power and add them. However the roots turn, the total is a whole number when the coefficients are, and Newton's identities produce it from the coefficients one step at a time — no root is ever found. Run the rule on x³ − x − 1 and out comes Perrin's sequence, whose terms know which numbers are prime, nearly.

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

Necklaces made of symmetries

Lagrange's theorem says a subgroup's size divides the group's, and the converse is false. One piece of the converse is true: every prime that divides the size is the order of some element. The proof threads the group's own elements onto a necklace whose product is nothing, turns it, and counts — the argument that proved Fermat's little theorem with beads, with the beads replaced by motions.

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.

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

Discs that fence in the eigenvalues

Draw one disc for each row of a square matrix, centred on the diagonal entry, with a radius equal to the sum of the sizes of everything else in that row. Every eigenvalue lies inside one of the discs, and a group of discs set apart from the rest holds exactly as many eigenvalues as it has discs. Nothing is solved to find them.

The highest point on the sphere is an eigenvalue

For a symmetric matrix, walk a unit arrow over every direction and record the value of xᵀAx. The highest value reached is the largest eigenvalue, the lowest is the smallest, and every eigenvalue in between is a saddle height, a minimum of maxima. From that one description comes a theorem no formula for the roots could give: delete a row and its column, and every eigenvalue of what is left sits between two of the original's.

What is left when the middle is taken out

Cut a finite piece out of a group's picture and count the parts of what remains that run off forever. The integers leave two, the plane one, a tree more with every cut — and no group anywhere leaves exactly three, because a third end is always the first of infinitely many.

The polygon a lattice becomes from far away

Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.

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.

Thirty-one moves from solved

Parity settles which half of a sliding puzzle's arrangements can be reached and is silent about how far away any of them is. Searching every reachable arrangement of the three-by-three tray answers the second question exactly — two arrangements sit thirty-one moves out — and parity turns up again, this time as a law about distance.

Which graphs let the tokens go anywhere

A sliding puzzle is a graph with a token on every vertex but one. Richard Wilson found in 1974 what every such puzzle can reach, and the answer has a surprise in it — the half the tray is stuck with is not a fact about permutations at all, but about the board being two-coloured — and one exception, a graph of seven vertices that reaches exactly 120 of 720.

The biggest box built from signs

Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.

The count a fold cannot change

A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.

Two graphs the eigenvalues cannot tell apart

A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.

A hole is a cycle that bounds nothing

A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.

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.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

Almost none of the roots are real

Pick the coefficients of a polynomial of degree a thousand at random, each one an independent draw from the bell curve, and ask how many of its thousand roots are real. The answer is about five. Mark Kac found in 1943 that the average grows only like (2/π) ln n, and the reason can be read off the real line itself: the real roots crowd towards +1 and −1 and spread evenly on a logarithmic scale of distance from them.