Fields & Galois theory

Field extensions, the Galois correspondence, the algebra↔geometry dictionary (Nullstellensatz), and impossibility-by-invariant — when something simply cannot be done.

Galois theory is the part of algebra that turns “can you reach this target with these tools?” into a question about symmetry — and answers it with a hard no when the symmetry says so. Its two great dualities — the Galois correspondence (fields ↔︎ symmetry groups) and the Nullstellensatz (equations ↔︎ solution sets) — are both Galois connections that snap shut into exact bijections precisely when a sharp side-condition holds. It is also the native algebra of engineering: the finite fields \(\mathrm{GF}(2^n)\) behind Reed–Solomon codes, AES, and CRCs live and breathe here. The headline is impossibility-by-invariant: the same machinery that proves you cannot trisect an angle with compass and straightedge is the template for proving this state is unreachable with this control law.

Fields, and the ladder of capability upgrades

A field is a number system where you can add, subtract, multiply, and divide by anything nonzero — \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\), and the finite field \(\mathbb{F}_p=\mathbb{Z}/p\) are canonical. Formally it is a commutative ring with \(1\ne 0\) in which every nonzero element is a unit, and it carries a characteristic — either \(0\) or a prime \(p\). (\(\mathbb{Z}\) is not a field: you cannot divide, so it is only a ring.)

The action begins when you enlarge a field. A field extension \(F/k\) (read “\(F\) over \(k\)”) is a bigger field \(F\) containing a smaller one \(k\) — a capability upgrade: \(\mathbb{R}(i)=\mathbb{C}\) adds a square root of \(-1\); \(\mathbb{Q}(\sqrt 2)\) adds a square root of \(2\); and \(\mathrm{GF}(2^8)/\mathrm{GF}(2)\) adds enough roots that AES can multiply bytes and Reed–Solomon can locate errors. Decisively, \(F\) is automatically a vector space over \(k\), so “how much bigger is \(F\)?” has a one-number answer.

• Degree \([F:k]=\dim_k F\) — the dimension of \(F\) as a \(k\)-vector space, a single integer (or \(\infty\)). This is the workhorse numerical invariant for every impossibility proof below.
• Tower law — for \(k\subseteq E\subseteq F\), degrees multiply: \[[F:k]=[F:E]\,[E:k].\] Each stage degree therefore divides the total. A prime total degree forbids any proper intermediate field.

Constrained vs free: algebraic elements and minimal polynomials

Inside an extension, each new element is one of two types. An element \(\alpha\in F\) is algebraic over \(k\) if it satisfies some nonzero polynomial equation with coefficients in \(k\) — it is constrained by an equation — and then \([k(\alpha):k]<\infty\). Otherwise \(\alpha\) is transcendental: utterly free, with \(k(\alpha)\cong k(x)\) the (infinite) field of rational functions. The number \(\sqrt 2\) is algebraic over \(\mathbb{Q}\) (it solves \(x^2-2=0\)); \(\pi\) is transcendental.

For an algebraic \(\alpha\) there is a single equation that says everything:

The two worlds: symmetry, fixed fields, and the Galois-ness gap

To get a correspondence we need a second world to pair against the fields. That world is symmetry. The automorphism group \(\mathrm{Aut}_k(F)\) — the Galois group when the extension is nice — consists of all field automorphisms of \(F\) that fix \(k\) pointwise, made into a group under composition. Its elements permute the roots of every \(k\)-polynomial: this is the “symmetry content” of the extension.

Running the construction backwards, any subgroup \(G\le\mathrm{Aut}(F)\) has a fixed field \(F^G\) — everything left unmoved by every symmetry in \(G\). This is a subfield sitting above \(k\), and it is antitone: a bigger group of symmetries fixes a smaller field. That order-reversal is the heart of the whole theory.

Two structural facts control how tightly the two worlds fit:

• Artin’s theorem. For a finite group \(G\) of automorphisms, \([F:F^G]=|G|\), and \(F/F^G\) is automatically a well-behaved (Galois) extension with group \(G\).
• The general inequality. Always \(|\mathrm{Aut}_k(F)|\le[F:k]\). Equality is special.

The Galois correspondence

Splitting fields are the raw material: the splitting field of a polynomial \(p\) is the smallest extension in which \(p\) factors completely into linear pieces (e.g. \(\mathbb{Q}(\sqrt 2)\) for \(x^2-2\)), and the algebraic closure \(\overline{k}\) — modelled by \(\mathbb{C}=\overline{\mathbb{R}}\) — is the smallest extension in which every polynomial splits. When \(F/k\) is Galois, the two worlds we built are not merely connected; they are the same lattice, read upside down.

Reading a tower of fields top-to-bottom corresponds to reading a chain of subgroups bottom-to-top — the order flips, intersections trade places with joins, and it is a genuine lattice anti-isomorphism. The simplest non-trivial picture is \(\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}\), whose group is \(G=\{1,\sigma,\tau,\sigma\tau\}\cong(\mathbb{Z}/2)^2\) (with \(\sigma:\sqrt2\mapsto-\sqrt2\), \(\tau:\sqrt3\mapsto-\sqrt3\)). The middle layer of three fields matches, in reverse, the three order-2 subgroups:

\[ \begin{CD} F=\mathbb{Q}(\sqrt2,\sqrt3) @>{\text{Aut}_{(-)}F}>> \{1\} \\ @AAA @VVV \\ \mathbb{Q}(\sqrt2) @>>> \langle\tau\rangle \\ @AAA @VVV \\ \mathbb{Q} @>>{F^{(-)}}> G \end{CD} \]

The left column climbs through fields (inclusions pointing up); the right column descends through the matched subgroups (inclusions pointing down). The horizontal arrows are the two inverse maps of the correspondence — passing from a field to the subgroup that fixes it, and back. The full lattice, written out, is:

Notice the columns of degrees and orders multiply to \(4=[F:\mathbb{Q}]=|G|\) on every row — the tower law and Lagrange’s theorem in lockstep.

The native engineering instance is even cleaner. Every extension \(\mathrm{GF}(p^n)/\mathrm{GF}(p)\) is Galois with a cyclic group generated by a single symmetry, the Frobenius automorphism \(\varphi:x\mapsto x^{p}\). Subfields then correspond to subgroups of \(\mathbb{Z}/n\) — i.e. to divisors of \(n\): \(\mathrm{GF}(p^d)\subseteq\mathrm{GF}(p^n)\) exactly when \(d\mid n\), matched to \(\langle\varphi^{d}\rangle\). Frobenius also names the conjugates: the roots of an irreducible factor are the cyclotomic coset \(\{\beta,\beta^{p},\beta^{p^2},\dots\}\), the orbit of \(\varphi\). This is precisely the bookkeeping behind BCH/Reed–Solomon code design, where one chooses roots by their conjugacy classes under \(x\mapsto x^p\).

The other duality: the Nullstellensatz dictionary

Galois theory has a geometric twin that translates algebra into geometry. Fix an algebraically closed field. Polynomial equations (packaged as an ideal \(I\)) and their solution sets (varieties) correspond, again antitonely, via \[V(I)=\{p: f(p)=0\ \ \forall f\in I\},\qquad I(S)=\{f: f|_S=0\}.\] Here \(V(I)\) is the feasible set cut out by the constraints \(I\), and \(I(S)\) is the set of constraints forced by a region \(S\). More constraints carve out a smaller feasible set: order-reversing, a Galois connection once more.

The radical \(\sqrt I\) identifies \(f=0\) with \(f^2=0\): it models a constraint satisfied robustly, the closest algebraic gesture toward “spec satisfied with margin.” The whole dictionary is functorial — a duality in the categorical sense.

Impossibility-by-invariant and solvability by radicals

Now the payoff — both dualities feed one methodology: a target is unreachable by a fixed set of operations if attaining it would violate a numerical invariant those operations cannot change. Enlarge the toolkit and the boundary moves, but never past the invariant.

The engineering reflex is the same. For a linear system \(\dot x=Ax+Bu\), no control input \(u\) can ever steer the state out of the reachable subspace \(\mathcal{R}=\operatorname{im}[\,B\ AB\ \cdots\ A^{n-1}B\,]\) — an \(A\)-invariant subspace is the obstruction, and a target outside it is unreachable with this actuation, full stop, just as \(\sqrt[3]{2}\) is unreachable with compass and straightedge. Adding actuators enlarges \(B\) and moves the boundary; it never lets you escape the invariant subspace the current \((A,B)\) defines.

The classical case is straightedge-and-compass construction. Every constructible number sits atop a tower of degree-\(2\) extensions, so \([\mathbb{Q}(\alpha):\mathbb{Q}]\) must be a power of \(2\) — one fact that kills three ancient problems at once:

The invariant doing all the work is the degree — a dimension invariant pressed into service as an obstruction. Swap in a marked ruler or paper-folding (origami), and the reachable degrees grow: the boundary shifts, exactly as enlarging an operation set should.

The deepest instance is solvability by radicals — the precise sense in which a “closed-form expression” exists, built from the coefficients by \(+,-,\times,\div\) and \(n\)-th roots. Adjoining an \(n\)-th root is, in symmetry terms, a single cyclic (abelian) step. Stacking such steps means the Galois group must decompose into abelian pieces:

Recap

• A field lets you divide; a field extension \(F/k\) is a capability upgrade measured by one number, the degree \([F:k]\), which multiplies along towers and constrains intermediates by divisibility (P3).
• The minimal polynomial is the canonical minimal invariant (P13) deciding an algebraic element’s reachability — always relative to the base field.
• The Galois correspondence is an order-reversing bijection between intermediate fields and subgroups, exact precisely when the extension is normal + separable; the Galois-ness gap measures how far a duality falls short and flags decoupling (P9, RD7).
• The Nullstellensatz is the twin duality, equations ↔︎ feasible sets (P5), valid over algebraically closed fields and for equalities only — a false friend for real, inequality-defined trade-off fronts.
• Impossibility-by-invariant is the unifying message: degree forbids three classical constructions; non-solvability of \(S_{n\ge5}\) forbids any closed-form quintic formula; an \(A\)-invariant subspace forbids steering a linear system off its reachable set. The materials analogue (P11, RD7) is a provable “no recipe in this family can hit this spec” — sharp only relative to the stated operation set.