The purpose is to present a counterexample to the class field tower problem (CFTP) that arises in elementary class field theory. Interestingly, the solution uses a lot of noncommutative algebra and has a different flavor than a lot of other algebraic number theory. The problem was posed in 1925 by Furtwängler and was only solved in 1964 by Golod and Shafarevich. In fact, their work, and the solution we present here, also provided the first counterexample to the famous Burnside problem. We broadly follow [Ers] which can be found online here.

For more background on pro-p groups, Golod-Shafarevich groups, completed group algebras, and the general context of the solution, I refer the reader to the above notes or my more detailed write up available upon request. We begin with some definitions from elementary algebraic number theory.

For any number field $K$, let $\mathcal I_K$ be the group of fractional ideals in $\mathcal O_K$ and $\mathcal P_K$ the subgroup of principal ideals. The finite quotient \begin{equation} \mathrm{Cl}_K:=\mathcal I_K/\mathcal P_K \end{equation}

is the ideal class group of the number field. This group is a really important invariant of the number field $K$: it’s order, called the class number of $K$, measures how far $\mathcal O_K$ is from being a PID.

Let $\mathbb{H}(K)$ denote the Hilbert class field of $K$; this is the maximal abelian unramified extension of $K$. Beautifully, by the formalism of class field theory, we have that \begin{equation} \mathrm{Gal}(\mathbb{H}(K)/K)\cong\mathrm{Cl}_K. \end{equation} This means that a number field has $\mathcal O_K$ a PID if and only if it has no abelian unramified extensions. By Minkowski’s bound, this gives an amusing proof of unique prime factorization in $\mathbb Z$. For more background on basic number theory and class field theory, see $\left[\text{Cox}\right]$.

Let’s now state the problem at hand. It has a slightly dynamical flavor that I find very appealing.

The Class Field Tower Problem (CFTP)

Let $K$ be a number field and consider the tower of extensions \begin{equation} K=\mathbb{H}^0(K)\subset \mathbb{H}^1(K)\subset \mathbb{H}^2(K)\subset \dots \end{equation} where $\mathbb{H}^n(K)=\mathbb{H}(\mathbb{H}^{n-1}(K)).$ We call the sequence ${\mathbb{H}^i(K)}_{i=0}^\infty$ the class field tower of $K$. The question is this: for any number field $K$, does the class field tower ${\mathbb{H}^i(K)}$ eventually stabilize? That is, does there exist some $N\in \mathbb{N}$ such that for every $i\geq N$, $[\mathbb{H}^i(K):\mathbb{H}^N(K)]=1$? There’s a nice equivalent formulation: given a number field $K$, does the exist a finite extension $L$ such that $\mathcal O_L$ is a PID?

Proof of equivalence: One direction is easy. If the CFTP has a positive solution, then for any number field $K$, we can find some \begin{equation} [\mathbb{H}^i(K):\mathbb{H}^{i-1}(K)]=1. \end{equation} Then since $\mathrm{Gal}(\mathbb{H}^i(K)/\mathbb{H}^{i-1}(K))\cong C_{\mathbb{H}^{i-1}(K)}$ are both trivial, we have $\lvert C_{\mathbb{H}^{i-1}(K)}\rvert =1$ and hence $\mathcal{O}_{\mathbb{H}^{i-1}(K)}$ is a PID.

Conversely, suppose we can find some finite extension $L/K$ such that $\lvert C_L\rvert =1$. Since the extensions $\mathbb{H}^i(K)/\mathbb{H}^{i-1}(K)$ are abelian and unramified for each $i$, so are the compositum extensions $L\mathbb{H}^i(K)/L\mathbb{H}^{i-1}(K)$. Since $\lvert C_L\rvert =1$, $\mathbb{H}(L)=L$ so any abelian unramified extension of $L$ must be trivial. We have \begin{equation} L=L\mathbb{H}^1(K)=L\mathbb{H}^0(K)=LK=L. \end{equation} Inductively, this gives $L\mathbb{H}^i(K)=L$ and hence $\mathbb{H}^i(K)\subset L$ for every $i\in \mathbb N$. Since $L$ is a finite extension, the class field tower must eventually stabilize. $\square$

The first formulation will be used to construct the counterexample, the second is nice motivation.

The counterexample is essentially constructed as follows: we isolate the pro-p subgroup of the limiting Galois group, show the corresponding $\mathbb F_p$-algebra is infinite dimensional, and conclude the group itself (and hence the extension) is infinite.

The Magnus algebra over a field k and a finite set of variables $U$, denoted $k\langle\langle U\rangle\rangle$ is the algebra of formal, non-commuting power series in the variables from $U$. For example, \begin{equation} x+xy+xy^2+xy^3+\dots\neq x+yx+y^2x+y^3x+\dots \end{equation} in $\mathbb C\langle\langle {x,y}\rangle\rangle$.

Some Definitions. Let $A=\mathbb F_p\langle\langle U\rangle\rangle/I$ where $I$ is some closed two-sided ideal which is the kernel of the map $\pi:\mathbb F_p\langle\langle U\rangle\rangle\twoheadrightarrow A$. Notice that $A$ is naturally filtered by $A_n=\pi(F_n)$, lifting the degree filtration on $\mathbb F_p\langle\langle U\rangle\rangle$. Algebras (such as $A$) isomorphic to quotients of the Magnus algebra are called complete filtered algebras. Let $R\subset I$ be a generating set of the ideal and define $r_n=\lvert {r\in R: \deg(r)=n}\rvert $. Since $r_n=A_n/A_{n+1}$ is a finite-dimensional space we may assume $r_n<\infty$ for each $n$. Let $a_n=\dim_{\mathbb F_p}A_n/A_{n+1}$ and define the formal power series (called the Hilbert series) associated to the presentation $(U,R)$ of $A$

\begin{equation} \mathrm{Hilb}{A}(t)=\sum{n=1}^\infty a_nt^n \quad \text{and} \quad H_R(t)=\sum_{n=1}^\infty r_nt^n. \end{equation}

Similarly, for a presentation $(X,R)$ of pro-p group $G$ define the series

\begin{equation} H_R^G(t)=\sum_{r\in R} t^{D(r)}. \end{equation}

The main tool for constructing the counterexample is the following theorem. We note that inequalities are viewed coefficient-wise as formal series.

Theorem 1. (Golod-Shafarevich Inequality for Complete Filtered Algebras)

Let $A=\mathbb F_p\langle\langle U\rangle\rangle/I$ be a complete filtered algebra and $R$ a generating set for the ideal $I$. Let the formal power series $H_R(t)$ and $\text{Hilb}_A(t)$ be as before. Then, \begin{equation} \frac{(1-\lvert U\rvert t+H_R(t))\cdot \text{Hilb}_A(t)}{1-t}\geq \frac{1}{1-t}. \end{equation}

I believe a proof has never been published in English, only in the original Russian paper [Vin]. At some point, I hope to transcribe the proof, at which point I will update this post.

We now have a few simple corollaries. The first and second give better criteria for when a complete filtered algebra is infinite. The third extends these results to pro-p groups by their completed group algebras.

Corollary 1. If we can find $\tau\in (0,1)$ such that $1-\lvert U\rvert \tau +H_R(\tau)\leq 0,$ then $\text{Hilb}_A(t)$ diverges and hence $A$ is infinite-dimensional.

Proof. Suppose we have $\tau\in (0,1)$ such that $1-\lvert U\rvert \tau+H_R(\tau)\leq 0$ but such that $\text{Hilb}_A(\tau)$ converges. Since $H_R(\tau)$ has non-negative coefficients, the above bound means it converges. Plugging into the Golod-Shafarevich inequality, this gives \begin{equation} \frac{(1-\lvert U\rvert \tau+H_R(\tau))\cdot \text{Hilb}_A(\tau)}{1-\tau}\geq \frac{1}{1-\tau} \end{equation} as an inequality of real numbers (not of power series as defined above). Hence, we obtain \begin{equation} (1-\lvert U\rvert \tau+H_R(\tau))\cdot \text{Hilb}_A(\tau)\geq 1. \end{equation} However, this is clearly a contradiction since $\text{Hilb}_A(\tau)>0$. $\square$

Corollary 2. Let $\mathcal A$ be a finite-dimensional complete filtered algebra with minimal presentation $(U,R)$ and assume $\lvert U\rvert \geq 2.$ By minimal, we mean that no proper subset of $U$ generates $\mathcal A.$ We have \begin{equation} \lvert R\rvert >\frac{\lvert U\rvert ^2}{4}. \end{equation}

Proof. If $\lvert R\rvert =\infty $ the statement is vacuously true since we always assume the generating set $U$ to be finite. Thus, without loss of generality, we can restrict to the case where $\lvert R\rvert <\infty$ and hence $H_R(t)<\infty$ is a finite sum. Recall the filtration ${\mathcal A_n}$ on $\mathcal A$. Since we assume $(U,R)$ to be minimal, we have $r_1=\lvert R\cap \mathcal A_1\rvert =0$ for otherwise one of the generators can be expressed as a power series of the others, contradicting the minimality of $(U,R).$ Now, for any $\tau >0$ we have $H_R(\tau)\leq \lvert R\rvert \tau^2$. Moreover, by Corollary 1 and the fact that $\mathcal A$ is assumed to be finite-dimensional, we have that $1-\lvert U\rvert \tau+H_R(\tau)>0$ giving \begin{equation} 0<1-\lvert U\rvert \tau +H_R(\tau)\leq 1-\lvert U\rvert \tau + \lvert R\rvert \tau^2. \end{equation} Thus, setting $\tau = 2/\lvert U\rvert $ gives \begin{equation} 1<\lvert R\rvert \frac{4}{U^2}\implies \lvert R\rvert >\frac{\lvert U\rvert ^2}{4}. \quad \square \end{equation}

Before the next result we need to introduce some notation. Let $d(G)$ be the minimal number of generators for $G$. That is, $d(G)=\lvert X\rvert$ where $G\cong\hat F_p(X)/N(R)$ and $\lvert X\rvert$ is minimal over all such presentations. In this context, let $r(G)=\lvert R\rvert.$ Let $\Phi(G)= \overline{[G,G]G^p}$ be the Frattini subgroup of $G$. The quotient

\begin{equation} G/\Phi(G)\cong (\mathbb{F}_p)^d \end{equation}

is an elementary $p$-group such that $d(G)=d(G/\Phi(G))=\mathrm{dim}_{\mathbb{F}_p}(G/\Phi(G))$.

Lemma. ([Ers, p. 13]) Let $G\cong \hat F_p(X)/N(R)$ be a pro-p group. Then $\lvert X\rvert =d(G)$ if and only if $R\subset \Phi(\hat F_p(X))$ if and only if $D(r)\geq 2 $ for all $r\in R.$

Corollary 3. ([Koc]) Let $G\cong \hat F_p(X)/N(R)$ be a pro-p group such that $X$ is minimal and finite. If $\lvert R\rvert \leq \lvert X\rvert ^2/4,$ then $G$ is infinite.

Proof. Suppose $G$ is a finite p-group. As before, if $\lvert R\rvert =\infty$ the claim is trivial so we may assume the presentation is finite. From the pair $(X,R)$ for $G$, we get the corresponding presentation $(U,{1-r:r\in R})$ for the completed group algebra $\mathbb F_p[[G]]$ with $\lvert U\rvert =\lvert X\rvert $ and with $\lvert {1-r: r\in R}\rvert = \lvert R\rvert $. Since $\mathbb F_p[[G]]$ is finite-dimensional, we have from Corollary 1 that

\begin{equation} 1-\lvert U\rvert t+H_R(t)>0 \end{equation} for all $t\in (0,1).$ Furthermore, since $X$ is assumed to be minimal, we have from the lemma that $\deg(1-r)\geq 2$ for all $r\in R$ yielding $H_R(t)\leq \lvert R\rvert t^2$ for $t>0.$ Hence, setting $t=2/\lvert X\rvert =2/\lvert U\rvert $ gives the result. $\square$

Constructing a Counterexample

Finally, we can find counterexamples giving a negative solution to the class field tower problem. Let $\mathbb H_p(K)$ be the maximal unramified extension of $K$ such that \begin{equation} \mathrm{Gal}(\mathbb H_p(K)/K) \cong (\mathbb Z/ p \mathbb Z)^d \end{equation}

is p-elementary. Construct the $p$-class field tower as before. Since every $p$-elementary group is in particular abelian, it follows that $\mathbb H_p(K)\subset \mathbb H(K)$. Hence, if the $p$-tower never stabilizes, neither does the standard tower.

For the rest of this section, let $G_{K,p}=\mathrm{Gal}(\mathbb H_p^\infty(K)/K)$ be the pro-p limit of the p-class field tower. If $G_{K,p}$ is infinite, since each step in the p-class field tower is a finite extension, this guarantees that the tower ${\mathbb H_p^i(K)}$ never stabilizes. The following number-theoretic theorem of Shafarevich gives us some information about the group $G_{K, p}$.

Theorem 2. ([Sha]) Let $K$ be a number field and $\nu(K)$ be the number of infinite primes of $K$. For any rational prime $p\in \mathbb Z$, we have

\begin{equation} 0\leq r(G_{K,p})-d(G_{K,p})\leq \nu(K)-1. \end{equation}

Since

\begin{equation} \mathrm{Gal}(\mathbb H_p(K)/K)\leq \mathrm{Gal}(\mathbb H(K)/K)\cong \mathrm{Cl}_K \end{equation}

is isomorphic to the maximal p-elementary subgroup of the class group, denoted $\mathrm{Cl}^p_K$, the above theorem says

\begin{equation} 0\leq r(G_{K,p})-\dim_{\mathbb F_p} \mathrm{Cl}_{p}^{K}\leq \nu(K)-1. \end{equation}

This allows us to prove the last result we need.

Theorem 3. ([Koc]) Assume that \begin{equation} \mathrm{dim}_{\mathbb{F}_p}\mathrm{Cl}_K^{p}>2+2\sqrt{\nu(K)+1}, \end{equation}

then $G_{K,p}$ is infinite so that the class field tower of $K$ never stabilizes.

Proof. To apply Corollary 3, we want \begin{equation} r(G_{K,p})\leq d(G_{K,p})^2/4= (\mathrm{dim}_{\mathbb{F}_p} \mathrm{Cl}^{p}_K)^2/4. \end{equation}

Using our assumption, we have \begin{equation} (\mathrm{dim}_{\mathbb{F}_p} \mathrm{Cl}^p_K-2)/2 > \sqrt{\nu(K)+1}, \end{equation}

and thus \begin{equation} (\mathrm{dim} \mathrm{Cl}^p_K(K))^2/4 > \nu(K)+{\dim \mathrm{Cl}^p_K}. \end{equation}

Applying the above theorem gives the result. $\square$

At last, we’ve reduced the problem to just finding a single number field $K$ satisfying the hypothesis of Theorem 3!

We fix $p=2$. For examples constructed over any prime, see [Ers, p.37]. Let $n>2+2\sqrt{2+1}\approx 5.4641$ and define \begin{equation} K=\mathbb Q(\sqrt{\varepsilon q_1\dots q_{n+1}}) \end{equation} where $q_i$ are distinct odd primes and $\varepsilon$ is chosen so that $\varepsilon q_1\dots q_{n+1}\equiv 1$ mod $4$. Breaking up the square root, let

\begin{equation} L=\mathbb Q(\sqrt{\varepsilon_1 q_1 }, \dots, \sqrt{\varepsilon_{n+1} q_{n+1}}) \end{equation}

where $\varepsilon_i$ is chosen so that $\varepsilon q_i \equiv 1$ mod $4$. Notice that the discriminant $\Delta_{L/K}=(\prod \epsilon_i q_i)^{2^{n-1}}$ is only divisible by $q_i$ and so the extension is unramified and that $\mathrm{Gal}(L/K)\cong (\mathbb Z/2\mathbb Z)^n$. Hence, $L\subset \mathbb{H}(K)$ and \begin{equation} \mathrm{Gal}(L/K)\cong (\mathbb Z/2\mathbb Z)^n \hookrightarrow \mathrm{Gal}(\mathbb H(K)/K) \end{equation} so that $\mathrm{dim}_{\mathbb F_p} \mathrm{Cl}^p_K \geq n.$ It’s obvious that $\nu(K)\leq \left[K:\mathbb Q\right]=2$.

In conclusion, we have \begin{equation} 2+2\sqrt{\nu(K)+1}\leq 2+2\sqrt{2+1} < n \leq \mathrm{dim}_{\mathbb F_p} \mathrm{Cl}^p_K. \end{equation} Applying Theorem 3 shows the field $K$ has an infinite class field tower. For example, the field (with $n=6$) \begin{equation} K=\mathbb{Q}(\sqrt{5\cdot 13 \cdot 17 \cdot 29 \cdot 37 \cdot 41 \cdot 53}) \end{equation} has an infinite class field tower.

References:

[Cox] Cox, D.A. (1989) Primes of the Form $x^2+ny^2$, Fermat, Class Field Theory and Complex Multiplication. Wiley, 432.

[Ers] Ershov, Golod-Shafarevich Groups: A Survey. International Journal of Algebra and Computation, (2012)

[Koc] H. Koch, Galois theory of p-extensions, Springer Monograbs in Mathematics, 2001.

[Sha] I. Shafarevich, Extensions with prescribed ramification points. (Russian) Inst. Hautes Etudes Sci. Publ. ´ Math. No. 18 (1963), 71-95

[Vin] E. B. Vinberg, On the theorem concerning the infinite-dimensionality of an associative algebra (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 209-214.