TANT 3 - Ostrowski's theorems

Hello there! These are notes for the third class of the course "Topics in algebra and number theory" held in Block 4 of the academic year 2017/18 at the University of Copenhagen.

In the previous lecture we have defined the set of places ΣK of a field K which can be divided into Archimedean and non-Archimedean places (and yes, the trivial place). Moreover we have seen as an example that if we have a Dedekind domain R and a prime ideal pR we can define a non-Archimedean absolute value ||p on the field K:=Frac(R) by setting |x|p:=cordp(x) for some constant cR>1, where ordp(x) is the maximum power of p which divides xR.

In this lecture we will see that this construction is actually almost a bijection when R=F[t] for some field F and R=OK for some number field K. The reason why this is not a bijection tout court is because on number fields we have the Archimedean absolute values given by embeddings KC and for function fields K=F(t) we have the absolute value || and we have absolute values which are not trivial on F.


To define this bijection we will need the following observation, which is part of the many strange things that happen in the non-Archimedean world.

 Remark 1  From now on all non-Archimedean absolute values are assumed to be non-trivial.

 Lemma 2  If ϕ:KR0 is a non-Archimedean absolute value then the closed unit ball Aϕ:={xKϕ(x)1} is a subring of K and the open unit ball mϕ:={xKϕ(x)<1} is its unique maximal ideal. Moreover A×ϕ=Aϕmϕ and Aϕ is a discrete valuation ring if and only if ϕ(K×)R>0 is a discrete subgroup.

 Proof   Clearly 0,1Aϕ and if x,yAϕ then xyAϕ. The key fact here is that the non-Archimedean property ϕ(x+y)max(ϕ(x),ϕ(y)) tells us that if two elements x,yAϕ are in the unit ball then also their sum x+yAϕ is in the unit ball! It is also immediately clear that mϕ is an ideal, thanks again to the non-Archimedean property, and that Aϕ is a valuation ring, i.e. for every xK either xAϕ or x1Aϕ (and in particular K=Frac(Aϕ)). Think about how different this is from the unit balls of R and C for instance.

Observe now that if xA×ϕ this means that there exists yA×ϕ such that xy=1, which gives us that ϕ(x)ϕ(y)=1 which would be impossible if xmϕ, because in that case ϕ(x)<1 and ϕ(y)1. Thus we get that A×ϕAϕmϕ. Vice versa if xAϕmϕ then in particular x0 and thus there exists yK such that xy=1. But this implies that 1=ϕ(x)ϕ(y)=ϕ(y) because ϕ(x)=1, and thus we have that yAϕ and xA×ϕ (and also yA×ϕ).

We have hence proved that A×ϕ=Aϕmϕ, and this implies immediately that the quotient κϕ:=Aϕ/mϕ is a field. Moreover if nAϕ is a maximal ideal then A×An and thus nmϕ which implies that n=mϕ, i.e. that mϕ is the unique maximal ideal of Aϕ.

Suppose now that Aϕ is a discrete valuation ring. This implies that mϕ=πAϕ for some πAϕ. If we use multiple times the fact that Aϕ=A×ϕmϕ we get that Aϕ=A×ϕπAϕ=A×ϕπ(A×ϕπAϕ)==+j=0πjA×ϕ. Thus since Aϕ is a valuation ring we get that for every xK× there exists uA×ϕ and kZ such that x=uπk. This implies in particular that ϕ(x)=ϕ(u)ϕ(π)k=ϕ(π)k and thus that ϕ(K×)=ϕ(π)Z is a discrete subgroup of R>0.

Vice versa suppose that ϕ(K×) is a discrete subgroup of R>0. Then we know (see Exercise 3) that ϕ(K×)=αZ for some αR>0. In particular we can take α<1 and we can choose πK× such that ϕ(π)=α. Clearly πmϕ and if xK× we have that ϕ(x)=ϕ(π)k for some kZ, which implies that x=uπk for some uA×ϕ. To conclude we only need to observe that if xmϕ then clearly kN1, and thus mϕ=πAϕ. Q.E.D.

 Exercise 3  Prove that all every discrete subgroup of (R,+) is cyclic. Deduce as a corollary that every discrete subgroup of (R>0,) is cyclic. (Hint: for the first part, use an argument similar to the Euclidean algorithm which shows that every ideal of Z is principal. For the second part, think about a way of relating the two groups...)

 Example 4  Let R be a Dedekind domain with field of fractions K:=Frac(R) and let pR be a prime ideal. As we have seen in the previous lecture we have an absolute value ||p associated to p. It is not difficult to prove that in this case A||p is the localization of R at p, i.e. A||p=Rp:={xyKxR and yRp}. Indeed if xR then |x|p1 and if yRp then |y|p=1, which implies that |x/y|p1, i.e. that RpA||p.
On the other hand let αA||p and write αR=ab1 with a,bR coprime ideals. Observe that ordp(α)0 and thus we have that b is coprime to p. Since a=(α)b we have that a and b define the same ideal class in Cl(R). We want now to "get rid of" p in the denominator of α, and to do so we use the following Exercise 5 to take an ideal cR such that c is in the class of [a]1Cl(R) and is coprime to p. Since [a]=[b]Cl(R) and c[a] we get that ac and bc are principal ideals, generated by x and y respectively. To conclude we obtain that αR=ab1=(ac)(bc)1=xyR which implies (after modifying x by a unit if necessary) that α=x/y for xR and yRp, i.e. that αRp.
Observe finally that the maximal ideal m||p is equal to pRp.

 Exercise 5  Prove that for every Dedekind domain R, every ideal aR and every prime ideal pR there exists an ideal bR such that pb and ba1 is principal. (Hint: use the unique factorization of ideals and the Chinese remainder theorem for R).

We are now ready to introduce the two main theorems of this lecture. We will start with the "function field" case.

 Theorem 6  Let F be a field and let R:=F[t] and K:=Frac(R)=F(t). Then every absolute value on K which is trivial on F is either trivial tout court or equivalent to || or to an absolute value ||p for some prime ideal pF[t].

 Proof  Let ϕ:K=F(t)R0 be a non-trivial absolute value which is trivial on F. Then since {n1K}nZFF(t) we have that ϕ is non-Archimedean by applying the result proved in Exercise 6 of the previous lecture.

Suppose now that ϕ(t)1. In this case for every polynomial P(t)=nj=0ajtjF[t] we have that ϕ(P(t))maxj{ϕ(aj)ϕ(t)}1 because ϕ(aj)=1 by assumption and ϕ(t)1 as we have supposed at the beginning of this paragraph. Thus F[t]Aϕ and p:=F[t]mϕ is a prime ideal of F[t]. To conclude we want to show that ϕ is equivalent to ||p.

To do so observe first of all that F[t] is a principal ideal domain and thus in particular that p=p(t)F[t] for some irreducible p(t)F[t]. Thus we have that m||p=p(t)Rp. So, if we proceed as in the proof of Lemma 2 and we observe that K=Frac(Rp), we obtain that we can write every αK× as α=up(t)k for some uR×p and kZ. In particular k=ordp(α).
Observe now that we can write u=x/y with x,yRp, which implies that ϕ(u)=ϕ(x)/ϕ(y)=1/1=1. Thus we have that ϕ(α)=ϕ(p(t))ordp(α)=(1ϕ(p(t)))ordp(α) and thus we get that ϕ is equivalent to ||p, because ϕ(p(t))<1.

Suppose finally that ϕ(t)>1. Then we have that F[t1]Aϕ and we can define again p:=F[t1]mϕ. Observe that t1mϕ and thus p=t1F[t1]. Since we have that F(t)=Frac(F[t1]) we can proceed as before and prove that ϕ is equivalent to ||p. To conclude we only need to observe that for every polynomial f(t)=dj=0ajtjF[t] we have that f(t)=tddj=0adjtj which implies that ordp(f(t))=d=deg(f(t)) and so ||p=||. Q.E.D.

The main ingredient of the previous proof was the fact that the ring R=F[t] is a principal ideal domain, which allowed us to relate the absolute value ϕ that we started with and the absolute value ||p.

We want now to prove a similar statement when R=OK is the ring of integers of a number fields. If we try to adapt the previous proof to this case, we run into two obstacles:
  • first of all, it is not true anymore that every absolute value ϕ on K=Frac(R) is non-Archimedean. Indeed we have already seen in the previous lecture that every embedding KC gives rise to an Archimedean absolute value on K;
  • secondly, it is not true anymore that R=OK is a principal ideal domain: take for example K=Q(5). More specifically, we can measure how "far" the ring OK is from being a principal ideal domain by means of the class group Cl(OK), which is trivial precisely when OK is a principal ideal domain. It is not known at present whether there exist infinitely many number fields K such that Cl(OK) is trivial, and more generally there is a conjecture of Cohen and Lenstra which tries to explain the statistical behavior of these groups in families of number fields. If you are curious about this you can have a look at this paper by Goldfeld (concerning the characterization of imaginary quadratic fields of class number one), this master thesis of Nevin and this recent preprint of Bartel and Lenstra.
 Nevertheless, we will manage to overcome the second obstacle using our very own Lemma 2 that we proved before. To overcome the first obstacle we will need to wait until the following lecture!

 Theorem 7  Let K be an algebraic number field and let ϕ:KR0 be a non-Archimedean absolute value on K. Then ϕ is equivalent to ||p for some prime ideal pOK.

 Proof  Observe first of all that ϕ(n)=ϕ(1++1)ϕ(1)=1 for every nN (and thus for every nZ) because ϕ is non-Archimedean (this was the easy part of Exercise 6 of the previous lecture).

We want now to prove that OKAϕ. To do so observe that every xOK satisfies an equation of the form xn=n1j=0ajxj for a0,,an1Z. Thus we get that ϕ(x)n=ϕ(xn)=ϕ(n1j=0ajxj)maxj=0,,n1(ϕ(aj)ϕ(x)j)maxj=0,,n1(ϕ(x)j) which clearly implies that ϕ(x)1 and thus that OKAϕ.

Now as before we set p:=OKmϕ and we want to prove that ϕ is equivalent to ||p. Note first of all that by definition ||p is discrete and so by Lemma 2 we have that A||p=(OK)p is a discrete valuation ring with a principal maximal ideal m||p=p(OK)p=π(OK)p for some element π(OK)p.

Now we proceed exactly as before! Every element xK× can be written as x=uπk for some kZ and u(OK)×p and we have that ϕ(v)=1 for every v(OK)×p. Thus we have that ϕ(x)=(1/ϕ(π))ordp(x) for all xK×. Observe finally that ϕ(π)<1 because πpmϕ and thus ϕ is equivalent to ||p. Q.E.D.

We are thus left with the analysis of the set of Archimedean places ΣK, of a number field K. We already know from Example 10 of the previous lecture that every embedding KC induces an abolute value on K. We will prove in the following lecture that every possible Archimedean absolute value of K arises in this way!

Conclusions and references

In this lecture we managed to:
  • describe a way to associate a local ring Aϕ to every absolute value ϕ;
  • characterize up to equivalence the set of all absolute values on a function field F(t) which are trivial when restricted to F;
  • completely characterize the set ΣK of non-Archimedean places of a number field K.
The main reference is again Section 1 of these notes of Peter Stevenhagen. Moreover, you can have a look at:

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