Friday, December 27, 2019

Behaviour of conjugacy by reduction modulo integers

Let $A$ and $B\in\mathrm{M}_n(\mathbf Z)$ be two square matrices with integer coefficients. Assume that they are conjugate by $\mathrm{GL}_n(\mathbf Z)$, namely, that there exists a matrix $P\in\mathrm{GL}_n(\mathbf Z)$ such that $B=P^{-1}AP$. Then we can reduce this relation modulo every integer $d\geq 2$ and obtain a similar relation between the images of $A$ and $B$ in $\mathrm M_n(\mathbf Z/d\mathbf Z)$.

Almost the same holds if $A$ and $B$ are only conjugate by $\mathrm{GL}_n(\mathbf Q)$, except for a few exceptions: we just need to take care to reduce the relation modulo integers $d$ that are coprime to the denominators of the coefficients of $P$ or of $P^{-1}$.

I was quite surprised at first to learn that the converse assertion is false. There are matrices $A$ and $B$
in $\mathrm M_2(\mathbf Z)$ whose images modulo every integer $d\geq 2$ are conjugate, but which are not conjugate by a matrix in $\mathrm{GL}_2(\mathbf Z)$.

An example is given by Peter Stebe in his paper “Conjugacy separability of groups of integer matrices”, Proc. of the AMS, 32 (1), mars 1972, p. 1—7.
Namely, set
\[ A = \begin{pmatrix} 188 & 275 \\ 121 & 177 \end{pmatrix} = \begin{pmatrix} 11\cdot 17+1 & 25\cdot 11 \\ 11^2 & 11\cdot 16+1 \end{pmatrix} \]
\[ B = \begin{pmatrix} 188 & 11 \\ 3025 & 177 \end{pmatrix} =
\begin{pmatrix} 11\cdot 17+1 & 11 \\ 11^2\cdot 25 & 11\cdot 16+1 \end{pmatrix}. \]
These matrices $A$ and $B$ have integer coefficients, their determinant is $1$, hence they belong to $\mathrm{SL}_2(\mathbf Z)$. They also have the same trace, hence the same characteristic polynomial, which is $T^2-365T+1$. The discriminant of this polynomial is $3\cdot 11^2\cdot 367$. This implies that their complex eigenvalues are distinct, hence these matrices are diagonalizable over $\mathbf C$, and are conjugate over $\mathbf C$.

In the same way, we see that they remain conjugate modulo every prime number $p$
that does not divide the discriminant. Modulo $3$ and $11$, we check that both matrices become
conjugate to $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$, while they become conjugate to $\begin{pmatrix}-1 & 1 \\ 0 & -1 \end{pmatrix}$ modulo $367$.

It is a bit more delicate to prove that if we reduce modulo any integer $d\geq 2$, then $A$ and $B$ become conjugate under $\mathrm{SL}_2(\mathbf Z/d\mathbf Z)$. Stebe's argument runs in two steps.
He first computes the set of matrices $V$ that conjugate $A$ to $B$, namely he solves the equation $VA=BV$. The answer is given by
\[ V=V(x,y) = \begin{pmatrix} x & y \\ 11 y & 25 x-y \end{pmatrix}. \]
Moreover, one has
\[ \det(V(x,y))=25x^2 - xy -11y^2. \]
Consequently, to prove that $A$ and $B$ are conjugate in $\mathrm{SL}_2(\mathbf Z/d\mathbf Z)$, it suffices to find $x,y\in\mathbf Z/d\mathbf Z$ such that $\det(V(x,y))=1 \pmod d$.
To prove that they are conjugate by $\mathrm{SL}_2(\mathbf Z)$, we need to find $x,y\in\mathbf Z$ such that $\det(V(x,y))=1$, and if we agree to be content with a conjugacy by $\mathrm{GL}_2(\mathbf Z)$, then solutions of $\det(V(x,y))=-1$ are also admissible.

Let us first start with the equations modulo $d$. By the Chinese remainder theorem, we may assume that $d=p^m$ is a power of a prime number $p$. Now, if $p\neq 5$, we can take $y=0$ and $x$ such that $5x=1\pmod {p^m}$. If $p=5$, we take $x=0$ and we solve $y$ for $-11y^2=1\pmod {5^m}$, which is possible since $-11\equiv 4\pmod 5$ is a square, and it is easy, by induction (anyway, this is an instance of Hensel's lemma), to produce $y$ modulo $5^m$ such that $y\equiv 3\pmod 5$ and $-11y^2=1\pmod{5^m}$.

On the other hand, the equation $25x^2-xy-11y^2=\pm 1$ has no solutions in integers.
The case of $-1$ is easy by reduction modulo $3$: it becomes $x^2+2xy+y^2=2$, which has no solution since $x^2+2xy+y^2=(x+y)^2$ and $2$ is not a square modulo $3$.
The case of $+1$ is rather more difficult. Stebe treats it by reducing to the Pell equation $u^2=1101y^2+1$ and shows by analysing the minimal solution to this Pell equation that $y$ is divisible by $5$, which is incompatible with the initial equation.

From a more elaborate point of view, $V$ is a smooth scheme over $\mathbf Z$ that violates the integral Hasse principle. In fact, $V$ is a torsor under the centralizer of $A$, which is a torus, and the obstruction has been studied by Colliot-Thélène and Xu, precisely in this context. However, I did not make the calculations that could use their work to reprove Stebe's theorem.

Tuesday, July 2, 2019

Irreducibility of cyclotomic polynomials

For every integer $n\geq 1$, the $n$th cyclotomic polynomial $\Phi_n$ is the monic polynomial whose complex roots are the primitive $n$th roots of unity. A priori, this is a polynomial with complex coefficients, but since every $n$th root of unity is a primitive $d$th root of unity, for a unique divisor $d$ of $n$, one has the relation
\[ T^n-1 = \prod_{d\mid n} \Phi_d(T), \]
which implies, by induction and euclidean divisions, that $\Phi_n \in \mathbf Z[T]$ for every $n$.
The degree of the polynomial $\Phi_n$ is $\phi(n)$, the Euler indicator, number of units in $\mathbf Z/n\mathbf Z$, or number of integers in $\{0,1,\dots,n-1\}$ which are prime to $n$.

The goal of this note is to explain a few proofs that these polynomials are irreducible in $\mathbf Q[T]$ — or equivalently, in view of Gauss's lemma, in $\mathbf Z[T]$. This also amounts to saying that $\deg(\Phi_n)=\phi(n)$ or that the cyclotomic extension has degree $\phi(n)$, or that the canonical group homomorphism from the Galois group of $\mathbf Q(\zeta_n)$ to $(\mathbf Z/n\mathbf Z)^\times$ is an isomorphism.

1. The case where $n=p$ is a prime number.

One has $T^p-1=(T-1)(T^{p+1}+\dots+1)$, hence $\Phi_p=T^{p-1}+\dots+1$. If one reduces it modulo $p$, one finds $\Phi_p(T)\equiv (T-1)^{p-1}$, because $(T-1)\Phi_p(T)=T^p-1\equiv (T-1)^p$. Moreover, $\Phi_p(1)=p$ is not a multiple of $p^2$. By the Eisenstein criterion (after a change of variables $T=1+U$, if one prefers), the polynomial $\Phi_p$ is irreducible.

This argument also works when $n=p^e$ is a power of a prime number. Indeed, since a complex number $\alpha$ is a primitive $p^e$th root of unity if and only if $\alpha^{p^{e-1}}$ is a primitive $p$th root of unity, one has $\Phi_{p^e}= \Phi_p(T^{p^{e-1}})$. Then the Eisenstein criterion gives the result.

Comment.From the point of view of algebraic number theory, this proof makes use of the fact that the cyclotomic extension $\mathbf Q(\zeta_p)$ is totally ramified at $p$, of ramification index $p-1$.
Consequently, it must have degree $p-1$. More generally, it will prove that $\Phi_p$ is irreducible over the field $\mathbf Q_p$ of $p$-adic numbers, or even over any unramified extension of it, or even over any algebraic extension of $\mathbf Q_p$ for which the ramification index is prime to $p-1$.

2. The classical proof

Let us explain a proof that works for all integer $n$. Let $\alpha$ be a primitive $n$th root of unity, and let $P\in\mathbf Z[T]$ be its minimal polynomial — one has $P\mid \Phi_n$ in $\mathbf Z[T]$. Let (A priori, the divisibility is in $\mathbf Q[T]$, but Gauss's lemma implies that it holds in $\mathbf Z[T]$ as well.) Fix a polynomial $Q\in\mathbf Z[T]$ such that $\Phi_n=PQ$.

By euclidean division, one sees that the set $\mathbf Z[\alpha]$ of complex numbers of the form $S(\alpha)$, for $S\in\mathbf Z[T]$, is a free abelian group of rank $\deg(P)$, with basis $1,\alpha,\dots,\alpha^{\deg(P)-1}$.

Let $p$ be a prime number which does not divide $n$. By Fermat's little theorem, one has $P(T)^p \equiv P(T^p) \pmod p$, so that there exists $P_1\in\mathbf Z[T]$ such that $P(X)^p-P(X^p)=pP_1(T)$. This implies that $P(\alpha^p)=p P_1(\alpha)\in p\mathbf Z[\alpha]$.

Since $p$ is prime to $n$, $\alpha^p$ is a primitive $n$th root of unity, hence $\Phi_n(\alpha^p)=0$. Assume that $P(\alpha^p)\neq 0$. Then one has $Q(\alpha^p)=0$. Differentiating the equality $\Phi_n=PQ$, one gets $nT^{n}=T\Phi'_n(T)=TP'Q+TPQ'$; let us evaluate this at $\alpha_p$, we obtain $n=\alpha^p P(\alpha_p) Q'(\alpha^p)=p \alpha^p P^1(\alpha^p)Q'(\alpha^p)$. In other words, $n\in p\mathbf Z[\alpha]$, which is absurd because $n$ does not divide $p$. Consequently, $P(\alpha^p)=0$, and $P$ is also the minimal polynomial of $\alpha^p$.

By induction, one has $P(\alpha^m)=0$ for every integer $m$ which is prime to $n$. All primitive $n$th roots of unity are roots of $P$ and $\deg(P)=\phi(n)=\deg(\Phi_n)$. This shows that $P=\Phi_n$.

Comment.Since this proof considers prime numbers $p$ which do not divide $n$, it makes implicit use of the fact that the cyclotomic extension is unramified away from primes dividing $n$. The differentiation that appears in the proof is a way of proving this non-ramification: if $P(\alpha^p)$ is zero modulo $p$, it must be zero.

3. Landau's proof

A 1929 paper by Landau gives a variant of this classical proof which I just learnt from Milne's notes on Galois theory and which I find significantly easier.

We start as previously, $\alpha$ being a primitive $n$th root of unity and $P\in\mathbf Z[T]$ being its minimal polynomial.

Let us consider, when $k$ varies, the elements $P(\alpha^k)$ of $\mathbf Z[\alpha]$. There are finitely many of them, since this sequence is $n$-periodic, so that they can be written as finitely polynomials of degree $<\deg(P)$ in $\alpha$. Let $A$ be an upper-bound for their coefficients. If $p$ is a prime number, we have $P(\alpha^p) \in p\mathbf Z[\alpha]$ (by an already given argument). This implies $P(\alpha^p)=0$ if $p>A$.

By induction, one has $P(\alpha^m)=0$ for any integer $m$ whose prime factors are all $>A$.

One the other hand, if $m$ is an integer prime to $n$ and $P$ is the product of all prime number $p$ such that $p\leq A$ and $p$ does not divide $m$, then $m'=m+nP $ is another integer all of which prime factors are $>A$. (Indeed, if $p\leq A$, then either $p\mid m$ in
which case $p\nmid nP$ so that then $p\nmid m'$, or $p\nmid m$ in which case $p\mid nP$ so that $p\nmid m'$.) Since $m'\equiv m \pmod n$, one has $P(\alpha^{m})=P(\alpha^{m'})=0$.

This shows that all primitive $n$th roots of unity are roots of $P$, hence $P=\Phi_n$.

Comment. —This proof is quite of a mysterious nature to me.

4. Using Galois theory to pass from local information to global information

The cyclotomic extension $K_n$ contains, as subextension, the cyclotomic extensions $K_{p^e}$, where $n=\prod p_i^{e_i}$ is the decomposition of $n$ has a product of powers of prime numbers. By the first case, $K_{p^e}$ has degree $\phi(p^e)=p^{e-1}(p-1)$ over $\mathbf Q$. To prove that $\Phi_n$ is irreducible, it suffices to prove that these extensions are linearly disjoint, which is the object of the following lemma.

Lemma. — Let $m$ and $n$ be integers and let $d$ be their gcd. Then $K_m\cap K_n=K_d$.

This is an application of Galois theory (and the result holds for every ground field as soon as its characteristic does not divide $m$ and $n$). Let $M$ be the least common multiple of $m$ and $n$. One has $K_N=K_m\cdot K_n$, and the cyclotomic character furnishes a group morphism $\operatorname{Gal}(K_N/\mathbf Q)\to (\mathbf Z/N\mathbf Z)^\times$. The Galois groups $\operatorname{Gal}(K_N/K_m)$ and $\operatorname{Gal}(K_N/K_n)$ corresponding to the subfields $K_m$ and $K_n$ are the kernels of the composition of the cyclotomic character with the projections to $(\mathbf Z/m\mathbf Z)^\times$ and $(\mathbf Z/n\mathbf Z)^\times$, and their intersection to the subgroup generated by these two kernels, which is none but the kernel of the composition of the cyclotomic character with the projection to $(\mathbf Z/d\mathbf Z)^\times$.

Sunday, May 19, 2019

Designs, Skolem sequences, and partitions of integers

Recently, my father offered me the first volume of a graphic novel by Jean-François Kierzkowski and Marek called La suite de Skolem — Skolem's sequences. I knew about the norwegian mathematician Thoralf Skolem for two different reasons (the Löwenheim-Skolem theorem in model theory, and some diophantine equations that Laurent Moret-Bailly put in a more geometric setting — see his series of papers on Problèmes de Skolem), but I had never heared about Skolem sequences.

They appear in his 1957 paper, On certain distributions of integers in pairs with given differences (Math Scand., 5, 57-68).
The question is to put the integers $1,2,\dots,2n$ in $n$ pairs $(a_1,b_1),\dots,(a_n,b_n)$ such that the differences are all different, namely $b_i-a_i=i$ for $i\in\{1,\dots,r\}$. One can put it differently: write a sequence of $2n$ integers, where each of the integers from $1$ to $n$ appear exactly twice, the two $1$s being at distance $1$, the two $2$s at distance $2$, etc.
For example, $4,2,3,2,4,3,1,1$ is a Skolem sequence of length $n$, corresponding to the pairs $(7,8), (2,4), (3,6),(1,5)$.

Le jeu des cavaliers (Jessica Stockholder) — photo V. Pantaloni
Le jeu des cavaliers (photo V. Pantaloni)
The possibility of such sequences has been materialized under the form of a giant sculpture Le jeu des cavaliers by Jessica Stockholder at the Institut des Hautes Études Scientifiques (IHÉS) at Bures sur Yvette.

There is a basic necessary and sufficient condition for such a sequence to exist, namely $n$ has to be congruent to $0$ or $1$ modulo $4$. The proof of necessity is easy (attributed by Skolem to Bang): one has $\sum_{i=1}^n(b_i-a_i)=n(n+1)/2$, and $\sum_{i=1}^n (b_i+a_i)=2n(2n+1)/2$, so that $\sum_{i=1}^n b_i=n(n+1)/4+2n(2n+1)/2=n(5n+3)/4$. If $n$ is even, this forces $n\equiv 0 \pmod 4$, while if $n$ is odd, this forces $5n+3\equiv 0\pmod 4$, hence $n\equiv 1\pmod 4$. The proof of  existence consists in an explicit example of such a sequence, which is written down in Skolem's paper.

Skolem's motivation is only alluded to in that paper, but he explains it a bit further next year. In Some Remarks on the Triple Systems of Steiner, he gives the recipe that furnishes such a system from a Skolem sequence. Steiner triple systems on a set $S$ is the datum of triplets of elements of $S$ such that each pair of two elements of $S$ appears exactly once. In other words, they are a $(3,2,1)$-design on $S$ — a $(m,p,q)$-design on a set $S$ being a family of $m$-subsets of $S$ such that each $q$-subset appears in exactly $p$ of those subsets. Some relatively obvious divisibility conditions can be written down that give a necessary condition for the existence of designs with given parameters, but actual existence is much more difficult. In fact, it has been shown only recently by Peter Keevash that these necessary conditions are sufficient, provided the cardinality of the set $S$ is large enough, see Gil Kalai's talk Designs exist! at the Bourbaki Seminar.

In the case of Steiner triple systems, the condition is that the number $s$ of elements of $S$ be congruent to $1$ or $3$ modulo $6$. Indeed, there are $s(s-1)/2$ pairs of elements of $S$, and each 3-subset of the triple system features 3 such pairs, so that there are $N=s(s-1)/6$ triples. On the other hand, each element of $S$ appears exactly $3N/s$ times, so that $(s-1)/2$ is an integer. So $s$ has to be odd, and either $3$ divides $s$ (in which case $s\equiv 3\pmod 6$) or $s\equiv 1\pmod 6$.

And Skolem's observation is that a family of $n$ pairs $(a_i,b_i)$ as above furnishes a triple system on the set $S=\mathbf Z/(6n+1)\mathbf Z$, namely the triples $(i,i+j,i+b_j+n)$ where $1\leq i,j\leq n$, thus constructing Steiner triple systems on a set whose cardinality $6n+1$, when $n\equiv 0,1\pmod 4$.

My surprise came at the reading of the rest of Skolem's 1957 paper, because I knew the result he then described but had no idea it was due to him. (In fact, it was one of the first homework my math teacher Johan Yebbou gave to us when I was in classes préparatoires.) And since this result is very nice, let me tell you about it.

Theorem.Let $\alpha>1$ and $\beta>1$ be irrational real numbers such that $\alpha^{-1}+\beta^{-1}=1$. Then each strictly positive integer can be written either as $\lfloor \alpha n\rfloor$, or $\lfloor \beta n\rfloor$ for some integer $n\geq 1$, but not of both forms.

First of all, assume $N=\lfloor \alpha n\rfloor=\lfloor \beta m\rfloor$. Using that $\alpha,\beta$ are irrational, we thus write $N< \alpha n<N+1$ and $N<\beta m<N+1$. Dividing these inequalities by $\alpha$ and $\beta$ and adding them, we get $N<n+m<N+1$, since $\alpha^{-1}+\beta^{-1}=1$. This proves that any given integer can be written only of one of those two forms.

Since $\alpha^{-1}+\beta^{-1}=1$, one of $\alpha,\beta$ has to be $<2$. Assume that $1<\alpha<2$. The integers of the form $\lfloor \alpha n\rfloor$ form a strictly increasing sequence, and we want to show that any integer it avoids can be written $\lfloor \beta m\rfloor$.

Set $\gamma=\alpha-1$, so that $\beta=\alpha/(\alpha-1)=1+1/\gamma$. 

For every integer, we have $\lfloor \alpha(n+1)\rfloor = \lfloor \alpha n\rfloor + 1$ or $\lfloor \alpha(n+1)\rfloor=\lfloor \alpha n\rfloor+2$, so that if $\lfloor \alpha n\rfloor + 1$ is avoided, one has $\lfloor \alpha (n+1)\rfloor=\lfloor\alpha n\rfloor +2$.

Then, $\lfloor \alpha n\rfloor = n+\lfloor \gamma n\rfloor=n+k-1$, where $k=1+\lfloor \gamma n\rfloor$. The inequalities $k-1<\gamma n <k$ imply $k/\gamma - 1/\gamma< n<k/\gamma$. Moreover, $\lfloor \alpha(n+1)\rfloor=n+1+\lfloor \gamma(n+1)\rfloor=n+k+1$, so that $k+1<\gamma(n+1)<k+2$, hence $n>k/\gamma+1/\gamma-1>k/\gamma-1$. This proves that $n=\lfloor k/\gamma\rfloor$. Then, $\lfloor k\beta\rfloor=k+\lfloor k/\gamma\rfloor=k+n=\lfloor \alpha n\rfloor +1$.