\documentclass[11pt,a4paper]{article} \usepackage[margin=25mm]{geometry} \usepackage{fontspec} \setmainfont{TeX Gyre Pagella} \usepackage{amsmath,amssymb,mathtools} \usepackage{graphicx} \usepackage{xcolor} \usepackage[unicode,colorlinks=true,linkcolor=blue,urlcolor=blue]{hyperref} \usepackage{bookmark} \usepackage{enumitem} \setlist{itemsep=3pt,topsep=5pt} \setlength{\emergencystretch}{3em} \newcommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \newcommand{\passthrough}[1]{#1} \newcommand{\pandocbounded}[1]{#1} \makeatletter \def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} \def\maxheight{\ifdim\Gin@nat@height>0.75\textheight0.75\textheight\else\Gin@nat@height\fi} \makeatother \setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} \hypersetup{pdfauthor={OpenAI GPT-6.1 Sol, Codex, Ultra effort}} \begin{document} \section{Profinite groups and infinite Galois theory}\label{profinite-groups-and-infinite-galois-theory} \emph{Written by OpenAI GPT-6.1 Sol in Codex, Ultra effort, October 2026. Self-checked by the writing AI; no independent review is claimed. Public domain (CC0).} An automorphism of an infinite algebraic extension is determined by its actions on finite subextensions. These actions cannot be chosen independently: restricting an action from a larger field must recover the action already chosen on a smaller one. This compatibility turns an infinite Galois group into a compact topological group. Class field theory will describe its abelian quotients by arithmetic data. We use field operations, polynomial division, finite-dimensional linear algebra and the basic definitions of compact Hausdorff spaces and product topology. Section 0 proves the finite Galois correspondence and the Chinese remainder theorem used below. We also prove the required compactness and multiquadratic facts. The irreducibility and finite Galois groups of cyclotomic polynomials are proved in \href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/cyclotomic-polynomials-and-their-automorphisms.html}{Cyclotomic fields}, Theorem 12.1. Here we use that theorem to pass to infinite extensions. Freely accessible comparisons are Milne's \emph{Fields and Galois Theory} and Kedlaya's \emph{Notes on class field theory}. \textbf{Prerequisite proof availability.} The named results below identify specific programme lessons. The \href{https://kokunoyumeto.github.io/open-math-courses/courses/NT-CFT/proof-dependencies.html\#lesson-1}{prerequisite record} distinguishes published proofs, supplied owner texts awaiting publication, and missing full proofs. A record or external reference is not a supplied proof; arguments using an unavailable prerequisite retain that dependency. \subsection{0. The finite algebra behind the limits}\label{the-finite-algebra-behind-the-limits} We first establish the algebraic results needed to recover an infinite field from its finite pieces. \textbf{Finite automorphism lemma.} If a finite group \(H\) of distinct automorphisms acts on a field \(E\), and \(F=E^H\), then \([E:F]=|H|\). Moreover \(E/F\) is separable and normal, and \(H\) is its full automorphism group. \textbf{Proof.} Put \(n=|H|\). Given \(n+1\) elements \(x_1,\ldots,x_{n+1}\) of \(E\), form the matrix with entries \(h(x_j)\), indexed by \(h\in H\). Its homogeneous system has a nonzero solution over \(E\). Choose a solution \(c=(c_j)\) with least possible nonzero support and scale one nonzero coordinate to 1. Applying any \(h_0\in H\) to every equation permutes the rows, so \(h_0(c)\) is another solution. The difference \(h_0(c)-c\) has smaller support, because the normalized coordinate vanishes. Minimality forces this difference to be zero. Thus every \(c_j\) belongs to \(F\), and the identity row gives a nontrivial \(F\)-linear dependence among the \(x_j\). Consequently \(d=[E:F]\leq n\). Distinct field homomorphisms are linearly independent over the target field. Indeed, a nontrivial relation with least number of terms cannot have one term. If \(h_1\) and \(h_r\) occur, select \(y\) with \(h_1(y)\ne h_r(y)\). Subtract \(h_r(y)\) times the relation evaluated at \(x\) from the relation evaluated at \(yx\). This removes the last term and leaves the first nonzero, contradicting minimality. Each member of \(H\) is \(F\)-linear, while \(\operatorname{Hom}_F(E,E)\) has dimension \(d\) over \(E\): evaluation on an \(F\)-basis identifies it with \(E^d\). Hence \(n\leq d\), proving equality. For \(x\in E\), multiply \(X-z\) over the distinct members \(z\) of its \(H\)-orbit. The resulting polynomial is fixed coefficientwise by \(H\), hence belongs to \(F[X]\), and has distinct roots in \(E\). The minimal polynomial of \(x\) over \(F\) divides it. Thus that minimal polynomial is separable and splits in \(E\). This proves both separability and normality. Finally a finite extension of degree \(d\) has at most \(d\) embeddings into an algebraic closure. To prove the bound, choose finitely many generators and adjoin them successively; at each step an embedding has at most the minimal-polynomial degree many extensions, since the new generator must go to one of its roots. Multiplying these bounds gives the extension degree by the tower law. Since the \(n=d\) members of \(H\) already attain the bound, there are no other automorphisms. \(\square\) For a finite separable extension the same generator argument gives exactly the degree many embeddings: at each stage the transformed minimal polynomial has its full number of distinct roots in an algebraic closure, and sending the generator to any of them defines an extension of the embedding. If the extension is normal, all these embeddings preserve the field and are automorphisms. Preservation implies surjectivity because the image has the same finite dimension over the ground field. \textbf{Finite Galois correspondence.} Let \(E/K\) be finite, separable and normal, with group \(G\). Subgroups \(H\subset G\) correspond bijectively and in reverse inclusion order to intermediate fields, by \[ H\longmapsto E^H,\qquad M\longmapsto\operatorname{Gal}(E/M). \] The degree is \([E^H:K]=[G:H]\). Normal subgroups correspond to normal extensions of \(K\), with quotient group \(G/H\). \textbf{Proof.} The automorphism lemma gives \(\operatorname{Gal}(E/E^H)=H\) and \([E:E^H]=|H|\). For an intermediate field \(M\), the extension \(E/M\) is still separable and normal: the minimal polynomial over \(M\) of any element of \(E\) divides its split separable polynomial over \(K\). The embedding count gives \(|\operatorname{Gal}(E/M)|=[E:M]\). Its fixed field contains \(M\), and the automorphism lemma gives it the same relative degree as \(M\), so the fields are equal. The tower law proves the displayed degree formula. A \(K\)-embedding of \(M\) into an algebraic closure extends to an embedding of \(E\) by the successive root construction. Normality of \(E/K\) makes the extension an element of \(G\). It follows that \(M/K\) is normal exactly when \(M\) is preserved by every element of \(G\). The equality \(g(E^H)=E^{gHg^{-1}}\), and the bijection just proved, make this equivalent to normality of \(H\). In that case restriction maps \(G\) onto \(\operatorname{Gal}(M/K)\), with kernel \(H\), proving the quotient assertion. \(\square\) \textbf{Primitive element lemma.} A finite separable extension \(E/K\) is generated by one element. For infinite \(K\), write \(E=K(a_1,\ldots,a_r)\), and list its \(d=[E:K]\) embeddings into an algebraic closure, as counted above. For each pair of different embeddings, the linear polynomial \(\sum_j c_j(\sigma(a_j)-\tau(a_j))\) is nonzero. Their product is a nonzero polynomial in the \(c_j\). A nonzero polynomial over an extension of an infinite field cannot vanish on every tuple from that field: induction on the number of variables chooses values at which a nonzero coefficient stays nonzero, and then avoids finitely many roots in the last variable. Choose \(c_j\in K\) making this product nonzero. The element \(a=\sum_j c_ja_j\) then has \(d\) distinct images. Its minimal polynomial has degree at least \(d\), and at most \([E:K]=d\); thus \(K(a)=E\). For finite \(K\), the field \(E\) is finite. Its multiplicative group is cyclic. To prove this, let \(m\) be its exponent. For each prime power dividing \(m\), choose an element attaining that prime-power order by taking a suitable power of an element whose order contains it. The product of these elements has order \(m\), because they commute and their orders are pairwise coprime. Every nonzero element is a root of \(X^m-1\), so the polynomial root bound gives \(|E^\times|\leq m\). Since \(m\) divides the group order, equality follows. A generator of \(E^\times\) consequently generates \(E\) as a field. This proves the lemma in both cases. \textbf{Chinese remainder lemma.} For pairwise coprime positive integers \(n_1,\ldots,n_r\), reduction gives \[ \mathbf Z/(n_1\cdots n_r)\mathbf Z \simeq\prod_{j=1}^r\mathbf Z/n_j\mathbf Z. \] Put \(N=n_1\cdots n_r\). For each \(j\), Bezout's identity supplies integers \(b_j,c_j\) with \(b_j(N/n_j)+c_jn_j=1\). Then \(e_j=b_jN/n_j\) has residue 1 at \(n_j\) and 0 at the other moduli. Thus \(\sum_j a_je_j\) lifts any desired tuple of residues. An integer in the kernel is divisible by every \(n_j\), hence by their product, proving injectivity. The divisibility assertion follows successively from Bezout: if \(a,b\) are coprime and both divide \(x\), then \(b\) divides \(x/a\). Bezout itself follows by choosing the least positive integer in the ideal \((a,b)\) and applying integer division to show that it divides both generators. This also proves the construction without assuming the remainder lemma in advance. The same construction proves the \textbf{polynomial remainder lemma} used later for completions: if \(f_1,\ldots,f_r\in k[X]\) are pairwise coprime, then \[ k[X]/(f_1\cdots f_r)\simeq\prod_j k[X]/(f_j). \] Polynomial division shows that an ideal generated by two polynomials is generated by a polynomial of least nonzero degree: division by that polynomial leaves a smaller-degree member of the ideal, hence zero. Coprimality therefore gives a polynomial Bezout identity. Apply it to \(f_j\) and the product of the other factors to construct the same idempotent lifts as above. An element in the kernel is divisible by each factor, and successive Bezout identities make it divisible by the product. This proves surjectivity and injectivity over every field \(k\). \subsection{1. What compatibility means}\label{what-compatibility-means} For every positive integer \(n\), put \[ \widehat{\mathbf Z}=\varprojlim_n\mathbf Z/n\mathbf Z. \] The indexing order is divisibility. An element is a family \((a_n)_n\), with \(a_n\in\mathbf Z/n\mathbf Z\), satisfying \(a_m\bmod n=a_n\) whenever \(n\mid m\). Addition and multiplication are coordinatewise. Give each finite ring its discrete topology and the limit the subspace topology of their product. The map \(\mathbf Z\to\widehat{\mathbf Z}\) is injective: an integer divisible by every positive integer is zero. Its image is dense. Indeed, finitely many compatible congruence conditions are all satisfied by an integer representing the coordinate at the least common multiple of their moduli. Over \(\mathbf F_q\), restriction from \(\mathbf F_{q^m}\) to \(\mathbf F_{q^n}\), for \(n\mid m\), takes the exponent of the Frobenius automorphism modulo \(m\) to that exponent modulo \(n\). Thus compatible Frobenius exponents give exactly this inverse limit. Notice that the algebraic cyclic subgroup generated by Frobenius contains only integral exponents. Taking its closure adds all compatible profinite exponents. The same construction works for an inverse system of finite groups \(G_i\) with homomorphisms \(G_j\to G_i\). Its inverse limit consists of the compatible tuples in \(\prod_iG_i\). A \textbf{profinite group} is a topological group isomorphic to such an inverse limit. \begin{figure} \centering \pandocbounded{\includegraphics[keepaspectratio,alt={Three finite stages of the 2-adic inverse limit. Each residue modulo 8 maps to its residue modulo 4, then modulo 2. The highlighted compatible residues are 3, 3 and 1.}]{assets/compatible-residues.png}} \caption{Three finite stages of the 2-adic inverse limit. Each residue modulo 8 maps to its residue modulo 4, then modulo 2. The highlighted compatible residues are 3, 3 and 1.} \end{figure} \emph{Figure 1. Finite approximations to \(\mathbf Z_2\). Each edge is reduction of a residue. The highlighted branch is the image of the integer 3. Continuing the tree indefinitely gives \(\mathbf Z_2\); this finite drawing is a schematic of the compatibility condition.} \subsection{2. Finite quotients recognize compact groups}\label{finite-quotients-recognize-compact-groups} \subsubsection{Proposition 1.1. Characterization of profinite groups}\label{proposition-1.1.-characterization-of-profinite-groups} A topological group is profinite if and only if it is compact, Hausdorff and totally disconnected. In a profinite group, open normal subgroups form a neighborhood basis at the identity. Every open subgroup is closed and has finite index. Conversely, every closed subgroup of finite index is open. \textbf{Proof.} First, a product of finite discrete nonempty spaces is compact. To see this directly, take a family of closed subsets with the finite intersection property. The filter it generates extends to a maximal proper filter, or ultrafilter, by Zorn's lemma. In a finite partition, exactly one part belongs to an ultrafilter: two disjoint parts cannot both belong, and if no part belonged then all their complements would belong, giving an empty intersection. Apply this to each coordinate partition and choose its selected coordinate value. The tuple so obtained has every basic neighborhood in the ultrafilter. It belongs to each original closed set, because otherwise its open complement would contain such a neighborhood and contradict membership of that closed set in the filter. Thus every closed family with the finite intersection property has nonempty intersection, the equivalent closed-set formulation of compactness. A product containing an empty factor is empty and compact as well. Products of Hausdorff spaces are Hausdorff, since a differing coordinate separates two points. An inverse limit of finite discrete groups is a closed subset of their product: each compatibility condition is an equality of two continuous maps to a finite Hausdorff space. It is consequently compact and Hausdorff. Any two distinct tuples differ at a coordinate, and that coordinate separates them by a clopen set. A connected subset can therefore have only one point. Finite intersections of coordinate kernels form a basis of open normal subgroups. For the converse, we first justify the clopen neighborhoods that total disconnectedness supplies. In a compact Hausdorff space \(X\), let \(Q_x\) be the intersection of all clopen sets containing \(x\). This intersection is connected. Otherwise write it as two disjoint nonempty compact sets, with \(x\) in the first, and choose disjoint open neighborhoods \(U,V\) of those sets. Each point outside \(U\cup V\) is excluded by a clopen neighborhood of \(x\). Compactness gives finitely many such neighborhoods whose intersection \(C\) lies in \(U\cup V\). Both \(C\cap U\) and \(C\cap V\) are clopen in \(X\). The first contains \(x\) and excludes the second part of \(Q_x\), contradicting the definition of \(Q_x\). In a totally disconnected space it follows that \(Q_x=\{x\}\). Applying compactness once more to the complement of any neighborhood of \(x\) produces a clopen neighborhood inside it. Now let \(G\) be compact, Hausdorff and totally disconnected, and choose a clopen neighborhood \(C\) of 1 inside a prescribed neighborhood. Compactness of \(C\) and continuity of multiplication give an identity neighborhood \(V\) such that \(VC\subset C\). Replace \(V\) by a symmetric smaller neighborhood. For \(g\in V\), both \(gC\subset C\) and \(g^{-1}C\subset C\), so \(gC=C\). The stabilizer \[ H=\{g\in G:gC=C\} \] is a subgroup containing \(V\), hence open; also \(H\subset C\), since \(1\in C\). Its cosets form an open cover of the compact space \(G\), so there are finitely many. The core \(N=\bigcap_{g\in G}gHg^{-1}\) is an intersection of finitely many conjugates, and is an open normal subgroup inside \(C\). The natural map \[ G\longrightarrow\varprojlim_{N\text{ open normal}}G/N \] is injective because these \(N\) form a basis and \(G\) is Hausdorff. It is surjective: the cosets prescribed by a compatible tuple have the finite intersection property, and compactness supplies a point in their intersection. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Finally, an open subgroup has open cosets; its complement is a union of them, so it is closed. Compactness makes its index finite. If a subgroup is closed and has finite index, its complement is a finite union of closed cosets, so it is open. \(\square\) The word \emph{closed} in the last assertion matters. We will construct a subgroup of index 2 which is dense and proper. \subsection{3. The correspondence for an infinite extension}\label{the-correspondence-for-an-infinite-extension} Let \(L/K\) be algebraic, normal and separable. Every finite collection of elements of \(L\) lies in a finite Galois extension \(E/K\) contained in \(L\): take the splitting fields of their minimal polynomials and their compositum. Give \(G=\operatorname{Gal}(L/K)\) the \textbf{Krull topology}, with the groups \(\operatorname{Gal}(L/E)\), for such \(E\), as an identity neighborhood basis. Restriction induces a topological isomorphism \[ G\simeq\varprojlim_{E/K\text{ finite Galois in }L}\operatorname{Gal}(E/K). \] It is injective because the fields \(E\) exhaust \(L\). Conversely, a compatible family defines an automorphism on their union; its inverse is the family of inverse automorphisms. This also proves that the topology just defined is the inverse-limit topology. In particular, \(G\) is profinite. Here is the embedding extension argument in the infinite case. In an algebraic closure \(\Omega\) of \(K\), partially order embeddings of subfields containing an intermediate field \(M\), extending a specified \(K\)-embedding of \(M\), by extension of both domain and map. The union along a chain is an embedding of its union field. Zorn's lemma supplies a maximal embedding. If its domain omits \(x\in\Omega\), transform the minimal polynomial of \(x\) by that embedding and choose a root in \(\Omega\). Polynomial division identifies the adjoining field with the quotient by the minimal polynomial, so this choice extends the embedding to include \(x\), a contradiction. Its domain is therefore all of \(\Omega\). The image is algebraically closed: pull a polynomial's coefficients back to \(\Omega\), split it there, and apply the embedding to its roots. It contains \(K\), and every element of \(\Omega\) is algebraic over \(K\). Its polynomial over \(K\) already splits in this image, so that element belongs to the image. Hence the extended embedding is an automorphism of \(\Omega\). Normality makes it and its inverse preserve \(L\), so restriction gives an automorphism of \(L\). In particular, \(G\to\operatorname{Gal}(E/K)\) is surjective for every finite Galois \(E/K\) inside \(L\). \subsubsection{Theorem 1.2. Infinite Galois correspondence}\label{theorem-1.2.-infinite-galois-correspondence} The assignments \[ H\longmapsto L^H, \qquad M\longmapsto\operatorname{Gal}(L/M) \] are inverse, inclusion-reversing bijections between closed subgroups of \(G\) and fields \(K\subset M\subset L\). An open subgroup corresponds to a finite extension of \(K\), with \([M:K]=[G:H]\). A closed normal subgroup corresponds to a Galois extension of \(K\), and \[ \operatorname{Gal}(M/K)\simeq G/H \] as topological groups. For any subgroup \(H\), closed or otherwise, \[ \operatorname{Gal}(L/L^H)=\overline H. \] \textbf{Proof.} The group fixing \(M\) is closed: it is the intersection of the stabilizers of its elements, and each stabilizer is open and closed. If \(x\in L\setminus M\), the separable minimal polynomial of \(x\) over \(M\) has another root in \(L\), since \(L/M\) is normal. The embedding sending \(x\) to that root extends to an automorphism of \(L/M\). Thus the fixed field of \(\operatorname{Gal}(L/M)\) is exactly \(M\). The action on each element of \(L\) is locally constant, so \(H\) and \(\overline H\) have the same fixed field. If \(\sigma\notin\overline H\), choose finite Galois \(E/K\) such that the coset \(\sigma\operatorname{Gal}(L/E)\) misses \(H\). The restriction \(\sigma|_E\) is outside the image \(H_E\) of \(H\) in \(\operatorname{Gal}(E/K)\). Finite Galois theory supplies an element of \(E^{H_E}\) moved by \(\sigma\). That element lies in \(L^H\). This proves the last displayed equality and the bijection for closed subgroups. If \(H\) is open, it contains \(\operatorname{Gal}(L/E)\) for some finite Galois \(E/K\). Hence \(L^H\subset E\), and finite Galois theory computes its degree as \([G:H]\). Conversely, if \(M/K\) is finite, finitely many generators have an open common stabilizer, namely \(\operatorname{Gal}(L/M)\). Conjugation gives \(L^{\sigma H\sigma^{-1}}=\sigma(L^H)\). Thus \(H\) is normal exactly when \(M=L^H\) is stable under all \(K\)-automorphisms of \(L\). By the extension fact, stability is equivalent to normality of \(M/K\); separability is inherited. Restriction to \(M\) is then a continuous surjection with kernel \(H\). Its quotient isomorphism is a homeomorphism by compactness. \(\square\) For two intermediate fields, taking their compositum corresponds to intersecting their closed subgroups. Taking their intersection corresponds to the closure of the subgroup generated by the two subgroups. These follow directly from the inclusion-reversing bijection. \subsection{4. Frobenius and cyclotomic characters}\label{frobenius-and-cyclotomic-characters} Inside an algebraic closure of \(\mathbf F_q\), the roots of \(X^{q^n}-X\) form a field: raising to \(q^n\) preserves sums and products in characteristic \(p\), and also preserves inverses of nonzero roots. The derivative is \(-1\), so there are exactly \(q^n\) roots. This gives a field of degree \(n\) over \(\mathbf F_q\). Conversely, a degree-\(n\) extension has \(q^n\) elements and every nonzero element satisfies \(x^{q^n-1}=1\) by the finite-group order theorem. It is therefore exactly this root field. Its Frobenius has order \(n\): its \(n\)-th power is the identity, while a smaller positive power \(d