The lattice that runs the other way

A tower whose degrees multiply treated a field extension as a vector space and got a number out of it: the degree, which multiplies along a tower and settles three classical impossibilities by divisibility. A number is a coarse instrument. What it cannot say is which fields sit between the bottom and the top, or how they are arranged.

Galois’s answer is that the arrangement is already known, because it is the arrangement of the subgroups of a group — and the group is a finite object that can be written out on a page.

The field, and the symmetries it has #

Take the polynomial $x^3 - 2$. It has one real root, the cube root of two, and two complex ones obtained by turning that root by a third of a full circle each way.

The smallest field containing all three is generated by the real cube root together with a primitive cube root of unity, and it has degree six over the rationals: three for the cube root, doubled by the square root of $-3$ hiding in the root of unity.

A symmetry of this field is a way of rearranging its elements that respects addition and multiplication and leaves every rational number alone. Such a thing is completely determined by where it sends the three roots, because everything in the field is built from them by arithmetic; and it must send roots to roots, because a root satisfies an equation with rational coefficients that the symmetry preserves.

So each symmetry is a permutation of three objects, and there are at most six. There are exactly six, and the count matching the degree is not a coincidence — it is what it means for the field to be the splitting field of a polynomial with no repeated roots.

Fixing, and being fixed #

Now the two constructions that make the correspondence.

Given a subgroup of the symmetries, look at everything it leaves alone. That is a field — sums and products of fixed things are fixed — and it sits between the rationals and the top. Given a field in between, look at the symmetries that leave all of it alone. That is a subgroup.

Each construction reverses inclusions, and the reason is a sentence. A bigger subgroup has more symmetries to satisfy, so fewer things survive all of them: more symmetry means less fixed. A bigger intermediate field has more elements to leave alone, so fewer symmetries qualify.

The theorem is that the two constructions are inverse to each other. Start with a subgroup, take what it fixes, take the symmetries fixing that — and the original subgroup comes back. Start with an intermediate field and the same round trip returns it. The correspondence is therefore a bijection, and it is order-reversing, and that is the whole of the picture in the hero figure.

Reading the six #

The list is short enough to go through, and going through it is what makes the correspondence stop being abstract.

The trivial subgroup fixes everything, so it names the whole field, of degree six. This is the degenerate end of the correspondence and it is worth reading anyway: it says the only number fixed by no requirement at all is any number, which is what “the whole field” means.

The whole group fixes only the rationals — a number left alone by every symmetry is one the polynomial’s roots cannot see — so it names the bottom field, of degree one.

The subgroup of order three, the two rotations of the roots and the identity, moves every root and fixes their ratios. Those ratios are the cube roots of unity, so it names the field generated by a root of unity: degree two.

The three subgroups of order two each fix one root and swap the other two. Each names the field generated by the root it fixes: degree three. There are three of these, they are different fields, and they are indistinguishable from one another by any rational equation — which is exactly the statement that the polynomial’s three roots are algebraically alike.

In every row, the order of the subgroup times the degree of the field is six. That is the correspondence’s arithmetic half and it is checked in the figure on every one of the six: index equals degree, or, said the other way round, the number of symmetries given up is the number of dimensions gained.

Why an equation’s solvability is a question about a group #

The correspondence turns a question about fields into a question about subgroups, and the point of doing so is that subgroups can be enumerated.

Solving an equation by radicals means building a tower: adjoin a square root, then a cube root, then another root, until the solutions are in reach. Each step of that tower is an intermediate field, so a tower of radicals corresponds — upside down — to a chain of subgroups, each one inside the last, with each quotient as simple as taking a root is.

Whether such a chain exists is a finite question about a finite group. For the cube roots of two the answer is yes: the group of six has the subgroup of three inside it, the quotients have sizes two and three, and the classical formula for a cubic is the tower that corresponds. For the general equation of degree five the group is the symmetries of five things, its even half has no proper normal subgroup at all, and the chain cannot exist — so no formula does.

That is the argument in outline and it is the reason the correspondence matters more than any particular example. The cube that will not double settles an impossibility by a degree count; this settles a much larger class of impossibilities by a structure count, and the structure lives in a group of finite size.

What the correspondence needs #

The bijection above is not true of every field extension, and the hypotheses are the part most easily skipped.

The extension must be normal: every irreducible polynomial with a root inside must have all its roots inside. The field generated by the real cube root alone fails this — it contains one root of $x^3-2$ and not the other two — and its symmetry group is trivial, so the correspondence would have one subgroup facing three intermediate fields.

The extension must be separable: the polynomial must have no repeated roots. Over the rationals this is automatic, which is why it can be ignored in a first pass and why it cannot be ignored over a field of finitely many elements.

Together these two make the extension Galois, and the theorem is that for a Galois extension the correspondence is a bijection with the number of symmetries equal to the degree. Drop either hypothesis and the group is too small to see everything, and the picture collapses on one side only.

Which fields are Galois over the bottom #

There is one refinement worth carrying, because it is the correspondence’s second sentence.

A subgroup is normal — closed under conjugation by the whole group — exactly when the field it names is itself a Galois extension of the bottom, and the symmetry group of that smaller extension is the quotient. In the example, the subgroup of order three is normal, and the field it names, generated by a cube root of unity, is a perfectly good Galois extension of degree two with two symmetries.

The three subgroups of order two are not normal, and correspondingly the three cubic fields are not Galois over the rationals: each contains one root of $x^3-2$ and misses two. Conjugating one of those subgroups by a rotation gives another of them, which on the field side says the three cubic fields are carried into one another by symmetries of the top — they are conjugate, which is the same word doing the same work on both sides.

So the correspondence does not merely match up two lists. It matches up two vocabularies, and words that were invented independently on the two sides turn out to be translations of each other.

The correspondence at work: which numbers are constructible #

The correspondence is not only a classification. It answers questions, and the cleanest example is one this site has already met from the other side.

A number is reachable by compass and straightedge exactly when it sits in a tower of quadratic extensions, and by the degree argument that forces its degree over the rationals to be a power of two. The correspondence sharpens this into a statement about the group: the number is constructible exactly when the Galois group of its minimal polynomial’s splitting field has a chain of subgroups descending to the trivial one with every step of index two.

A group of order six has no such chain, because six is not a power of two — there is nowhere for a chain of index-two steps to end. So the cube root of two is not constructible, and the classical impossibility has been re-derived without mentioning the tower at all.

The same machinery answers the harder question about polygons. A regular polygon with $n$ sides is constructible exactly when the group of the field generated by an $n$th root of unity has a chain of index-two subgroups; that group is abelian and its order is $\varphi(n)$, and an abelian group has such a chain exactly when its order is a power of two. Hence Gauss’s criterion, which was found a generation before the correspondence existed and which the correspondence explains rather than merely confirms.

That is what a good correspondence does. It does not produce new answers to the questions that motivated it — those were already answered — it produces a reason the old answers had the shape they had, and then it answers questions nobody had thought to ask.

Where it fails, and what it needs #

Infinite extensions break the counting. The correspondence as stated needs the extension to be finite; for infinite Galois extensions the subgroups have to be restricted to the closed ones in a topology on the group, and the naive statement has more subgroups than fields.

The group must act on the whole extension. Everything above is about symmetries fixing the bottom field pointwise. Symmetries fixing it merely as a set are a larger collection and correspond to nothing in this picture.

And the lattice is only as informative as the group. For an extension whose group is cyclic of prime order there are exactly two subgroups and therefore no intermediate fields at all, and the correspondence, while true, says nothing anybody wanted to know. Its value scales with how complicated the group is, which is the reverse of the usual situation.

Where it came from #

Galois wrote the theory in 1830 and 1831, in memoirs that were lost, rejected, and lost again, and finally in a letter written the night before the duel that killed him at twenty. What he had was not the correspondence as stated here — the language of groups and fields did not exist — but the essential move: attach to an equation a group of permutations of its roots, and read the equation’s solvability off the group.

The modern formulation, with fields, automorphisms and a lattice on each side, is Artin’s, from lectures in the 1930s and 1940s. Artin’s reformulation did something specific and worth noting: it made the group the primary object and the polynomial incidental. Galois’s group was the group of a polynomial; Artin’s is the group of an extension, and polynomials appear only when somebody wants an example.

That change of emphasis is why the theory outgrew its origin. Almost nothing in modern algebraic number theory is about solving an equation, and almost all of it is about a correspondence between subgroups and subfields.

What the pictures cannot show #

The lattices drawn here have six nodes because the example was chosen to have six. Nothing about the picture indicates how quickly that grows: the splitting field of a general quintic has a group of a hundred and twenty elements with a hundred and fifty-six subgroups, and the corresponding diagram is not a figure.

The three fields generated by the three cube roots are drawn as three separate boxes, which is right, and they are drawn as distinguishable, which is a limitation of writing things down. Nothing rational distinguishes them; the labels $\sqrt[3]{2}$, $\omega\sqrt[3]{2}$ and $\omega^2\sqrt[3]{2}$ depend on a choice of which root is called real, and the correspondence is symmetric under permuting them.

And the connecting lines in the hero figure record a bijection rather than an inclusion. The inclusions within each lattice — which subgroup sits inside which, which field contains which — are the structure being reversed, and the figure shows the pairing rather than the order. The reversal is visible only in the labels: order rises down the left column and degree falls down the right.

The ladder from here #

Below: a tower whose degrees multiply, which supplies the degree the correspondence turns into an index, and the blocks a subgroup cuts out, where index and normality are defined. Sideways: the cube that will not double, an impossibility settled by the degree alone, and which polygons can be drawn, where the group behind the construction is abelian and the tower always exists. Above: the insolubility of the quintic, infinite Galois theory, and the correspondence for coverings, which is the same theorem in topology.

What is worth carrying away #

Two lattices that arise from unrelated definitions turn out to be one lattice seen from two sides, with the order reversed. That reversal is the tell: it says the two are related by fixing something, and fixing is always order-reversing, because asking for more to be fixed leaves fewer things doing the fixing.

The pattern recurs wherever a group acts. Subgroups against fixed sets, subgroups against covering spaces, ideals against varieties: in each case a bigger object on one side names a smaller one on the other, and in each case the reversal is the first hint that a correspondence is there to be found.