Theorem 2.3.1. Principal ideal theorem.
Let \(L\) be the Hilbert class field of the number field \(K\text{.}\) Then every ideal of \(K\) becomes principal in \(L\text{.}\)
Section 2.3 The principal ideal theorem
Reference.
Subsection Statement of the theorem
For a change, we’re going to prove something, although the proof will depend on the Artin reciprocity law which we haven’t proved. Or rather, we’re going to sketch a proof that you will get to fill in by doing the exercises. (Why should I have all the fun?)
The following theorem is due to Furtwängler, a student of Hilbert. (It’s also called the “capitulation” theorem, because the word “capitulate” was formerly used to mean “to become principal”. Etymology left to the reader.)
Proof.
This will follow by combining Theorem 2.3.8 (construction of the transfer homomorphism), Lemma 2.3.9 (implication that vanishing of the transfer homomorphism implies the desired result), and Theorem 2.3.10 (vanishing of the transfer homomorphism).
Example 2.3.2.
If \(K = \QQ(\sqrt{-5})\text{,}\) then \(L = \QQ(\sqrt{-5}, \sqrt{-1})\text{,}\) and the nonprincipal ideal class of \(K\) is represented by \((2, 1+\sqrt{-5})\text{,}\) which is generated by \(1+\sqrt{-1}\) in \(L\text{.}\)
Remark 2.3.3.
The property of making every ideal of \(K\) principal is in no way specific to the Hilbert class field. See Exercise 1.
Subsection First steps of the proof
The idea of the proof is to apply Artin reciprocity to reduce to a problem purely in finite group theory, which we then solve. To this end, let \(M\) be the Hilbert class field of \(L\text{;}\) then an ideal of \(L\) is principal if and only if its image under the Artin map \(J_L \to \Gal(M/L)\) is trivial. So our first step will be to give a purely group-theoretic description of the map \(V\colon \Gal(L/K) \to \Gal(M/L)\) corresponding to the extension homomorphism \(\Cl(K) \to \Cl(L)\) (i.e., making the diagram in Figure 2.3.4 commute, in which the horizontal arrows are Artin maps).
In order to proceed further, we must extract more information about the Galois groups in question.
- The extension \(M/K\) is unramified because \(M/L\) and \(L/K\) are. It is also Galois: its image under any element of \(\Gal(\overline{K}/K)\) is still an unramified abelian extension of \(L\) and so is contained in \(M\text{.}\)
- The maximal subextension of \(M/K\) which is abelian over \(K\) is equal to \(L\text{.}\)
Subsection Translation into group theory
Definition 2.3.5.
Given a finite group \(G\text{,}\) let \(G^{\ab}\) denote the maximal abelian quotient of \(G\text{;}\) that is, \(G^{\ab}\) is the quotient of \(G\) by its commutator subgroup \(G'\text{.}\) Then the previous discussion implies that \(\Gal(M/L)\) is the commutator subgroup of \(\Gal(M/K)\) and \(\Gal(M/K)^{\ab} = \Gal(L/K)\text{.}\) We may thus relabel Figure 2.3.4 as in Figure 2.3.6.
We now give the purely group-theoretic interpretation of the map \(V\) in Figure 2.3.6.
Definition 2.3.7.
Let \(G\) be a finite group and \(H\) a (not necessarily normal) subgroup. Let \(g_1, \dots, g_n\) be left coset representatives of \(H\) in \(G\text{:}\) that is, \(G = g_1H \cup \cdots \cup g_nH\text{.}\) For \(g \in G\text{,}\) put \(\phi(g) = g_i\) if \(g \in g_iH\) (i.e., \(g_i^{-1}g \in H\)). Put
\begin{equation*}
V(g) = \prod_{i=1}^n \phi(gg_i)^{-1}(gg_i)\text{;}
\end{equation*}
then \(V(g)\) always lands in \(H\text{.}\) (Note that \(\phi(gg_1), \dots, \phi(gg_n)\) is a permutation of \(g_1, \dots, g_n\text{.}\))
Now consider what happens when we compose the map \(g \mapsto V(g)\colon G \to H\) (which is not necessarily a homomorphism) with the projection \(H \to H^{\ab}\text{.}\) It will follow from Theorem 2.3.8 that the resulting map \(G \to H^{\ab}\) is a homomorphism which factors through \(G^{\ab}\text{.}\) The induced map \(V\colon G^{\ab} \to H^{\ab}\) is called the transfer map (in German “Verlagerung”, hence the use of the letter \(V\) in the notation).
Theorem 2.3.8.
With notation as in Definition 2.3.7, the map \(V\colon G \to H^{\ab}\) is a homomorphism; it does not depend on the choice of the \(g_i\text{;}\) and induces a homomorphism \(G^{\ab} \to H^{\ab}\) (i.e., kills commutators in \(G\)).
Proof.
See Exercise 2 and Exercise 3. Alternatively, one can derive the existence of the homomorphism from properties of homology of finite groups; see Exercise 2.
Setting aside the proof of Theorem 2.3.8 for the moment, let’s see that this does indeed give the correct map in Figure 2.3.6 when we take \(G = \Gal(M/K)\) and \(H = \Gal(M/L)\text{,}\) so that \(G/H = \Gal(L/K)\text{.}\) This amounts to computing what happens when we apply all of the maps starting with a prime \(\gothp\) of \(K\) at the top left of the diagram.
Choose a prime \(\gothq\) of \(L\) over \(\gothp\) and a prime \(\gothr\) of \(M\) over \(\gothq\text{,}\) let \(G_{\gothr} \subseteq G\) be the decomposition group of \(\gothr\) over \(K\) (i.e., the stabilizer of \(\gothr\) under the action of \(G\) on the primes above \(\gothp\)), and let \(g \in G_{\gothr}\) be the Frobenius of \(\gothr\text{.}\) Keep in mind that since \(G\) is not abelian, \(g\) depends on the choice of \(\gothr\text{,}\) not just on \(\gothq\text{;}\) that is, there’s no Artin map into \(G\text{.}\)
Let \(\gothq_1, \dots, \gothq_r\) be the primes of \(L\) above \(\gothp\text{;}\) then the image of \(\gothp\) in \(L\) is \(\prod_i \gothq_i\text{,}\) and the image of that product under the Artin map is \(\prod_i \Frob_{M/L}(\gothq_i)\text{.}\) To show that this equals \(V(g)\text{,}\) we make a careful choice of the coset representatives \(g_i\) in the definition of \(V\text{.}\) Namely, decompose \(G\) as a union of double cosets \(G_{\gothr} \tau_i H\text{.}\) Then the primes of \(L\) above \(\gothp\) correspond to these double cosets, where the double coset \(G_{\gothr} \tau_i H\) corresponds to \(L \cap \gothr^{\tau_i}\text{.}\) Let \(m\) be the order of \(\Frob_{L/K}(\gothp)\) and write \(G_{\gothr} \tau_i H = \tau_iH \cup g\tau_i H \cup \cdots \cup g^{m-1}\tau_i H\) for each \(i\text{;}\) we then use the elements \(g_{ij} = g^j \tau_i\) as the left coset representatives to define \(\phi\) and \(V\text{.}\) Thus the equality \(V(g) = \prod_i \Frob_{M/L}(\gothq_i)\) follows from the following lemma.
Lemma 2.3.9.
If \(L \cap \gothr^{\tau_i} = \gothq_i\text{,}\) then
\begin{equation*}
\Frob_{M/L}(\gothq_i) = \prod_{j=0}^{m-1} \phi(g g_{ij})^{-1} g g_{ij}\text{.}
\end{equation*}
Proof.
See Exercise 4.
Subsection The final group-theoretic ingredient
With this, Theorem 2.3.1 follows from the following fact.
Theorem 2.3.10.
Let \(G\) be a finite group and \(H\) its commutator subgroup. Then the transfer map \(V\colon G^{\ab} \to H^{\ab}\) is zero.
Proof.
See Exercise 7.
Remark 2.3.11.
The fact that Theorem 2.3.10 is so general means that we can easily obtain some extensions of Theorem 2.3.1. For example, it was observed by Iyanaga that if \(L\) is the ray class field of \(K\) of some modulus \(\gothm\text{,}\) and \(\gothm \gotho_L\) is the extension of this modulus to \(L\) (that is, extend the finite part and take all places of \(L\) above the infinite places in \(\gothm\)), then the induced map \(\Cl^{\gothm}(K) \to \Cl^{\gothm}(L)\) again vanishes.
Subsection Additional remarks
Remark 2.3.12.
One important qualification of Theorem 2.3.1 is that \(L\) need not itself have class number 1. This raises the question of whether every number field \(K\) admits a finite (not necessarily abelian) extension which has class number 1.
One approach to constructing such an extension would be to consider the class field tower over \(K\text{,}\) in which \(K_0 = K\) and for each positive integer \(i\text{,}\) \(K_i\) is the Hilbert class field of \(K_{i-1}\text{.}\) If this were to stabilize at some point \(K_i\text{,}\) then the latter would be a field of class number 1.
However, Golod and Shafarevich showed that in certain cases this sequence grows without bound, in which case \(K\) admits an infinite (solvable) unramified extension. For example, if \(K\) is an imaginary quadratic field in which at least six distinct finite places of \(\QQ\) ramify, or a real quadratic field in which at least eight distinct finite places of \(\QQ\) ramify, then the 2-part of the class field tower is unbounded. See [4], Chapter IX for a proof of this statement and some further discussion, and [57] for some more results plus a survey of recent literature.
In fact, it can be shown (see Exercise 8) that the finiteness of the class field tower is equivalent to the existence of an extension of \(K\) of class number 1, which a priori need not be solvable or even Galois! That is, if a number field \(K\) is contained in a number field \(L\) of class number 1, then we can always ensure that \(L/K\) is Galois and solvable. In particular, by the previous paragraph there exist number fields which admit no finite extensions of class number 1.
Remark 2.3.13.
Let \(K\) be a number field. Let \(M\) be the \(\gotho_K\)-submodule of \(K[x]\) consisting of integer-valued polynomials, meaning those that map \(\gotho_K\) into itself. The field \(K\) is said to be a Pólya field if \(M\) admits a basis consisting of polynomials of pairwise distinct degrees; such a basis is called a regular basis. Any field with trivial class group is a Pólya field, but not conversely. The terminology is due to Zantema [61], who showed among other things that every cyclotomic field is a Pólya field.
Using Theorem 2.3.1, Leriche showed that the Hilbert class field of any number field is a Pólya field (see [36], Corollary 3.2). In particular, every number field can be embedded into a Pólya field via an abelian extension, whereas it is unknown whether every number field can be embedded into a field of class number one (and Remark 2.3.12 shows that solvable extensions are definitely not enough).
Exercises Exercises
1.
Let \(K\) be a number field. Prove without using class field theory that there exists a finite extension \(L/K\) such that every prime of \(K\) becomes principal in \(L\text{.}\)
Hint.
Since \(\Cl(K)\) is finite, it suffices to fix a prime \(\gothp\) and produce \(L\) such that \(\gothp\) becomes principal in \(L\text{.}\) Again by the finiteness of \(\Cl(K)\text{,}\) there exists a positive integer \(a\) such that \(\gothp^a\) is principal; adjoin a suitable root of a generator.
2.
With notation as in Definition 2.3.7, prove that the map \(V \colon G \to H^{\ab}\) is independent of the choice of \(g_1,\dots,g_n\text{.}\)
Hint.
Independence of the order of the cosets follows because we are mapping into \(H^{\ab}\text{.}\) To check independence of the representatives of individual cosets, change one \(g_i\) to \(g_ih\) for some \(h \in H\text{;}\) this changes \(\phi(gg_i)^{-1} gg_i\) by a conjugation which has no effect in \(H^{\ab}\text{.}\)
3.
With notation as in Definition 2.3.7, prove that the map \(V \colon G \to H^{\ab}\) is a homomorphism which factors through \(G^{\ab}\text{.}\)
Hint.
For the homomorphism, rewrite \(V(g) V(g')\) as \(\prod_i \phi^{-1}(gg_i) gg_i \phi^{-1}(gg_j) g'g_j\) where \(j\) runs through the indices in a different order so that \(g_i = \phi(gg_j)\text{.}\) For the factorization through \(G^{\ab}\text{,}\) check directly from the formula that \(V(g)V(g') = V(g')V(g)\text{.}\)
4.
Prove Lemma 2.3.9.
Hint.
Note that \(\phi(gg_{ij}) = g_{i(j+1)}\) for \(j=0,\dots,m-2\text{,}\) so all of the terms in the product vanish except for \(\tau_i^{-1} g^m \tau_i\text{.}\) We directly verify that this is the desired Frobenius element by noting that the residue class degree of \(\gothq\) over \(\gothp\) equals \(m\text{.}\) See also [38], Proposition IV.5.9.
5.
Let \(H \subseteq G\) be an inclusion of finite groups. Let \(G'\) and \(H'\) be the commutator subgroups of \(G\) and \(H\text{.}\) Let \(\ZZ[G]\) be the group algebra of \(G\text{,}\) i.e., the (noncommutative) ring of formal linear combinations \(\sum_{g \in G} n_g [g]\) with \(n_g \in \ZZ\text{,}\) multiplied according to the rule that \([g][h] = [gh]\text{.}\) Let \(I_G \subset \ZZ[G]\) be the ideal of sums \(\sum_{g \in G} n_g[g]\) with \(\sum_{g \in G} n_g = 0\) (called the augmentation ideal; see Section 3.3). Let
\begin{equation*}
\delta\colon H/H' \to (I_{H}+I_GI_{H})/I_GI_{H}
\end{equation*}
be the homomorphism taking the class of \(h\) to the class of \([h]-1\text{.}\) Prove that \(\delta\) is an isomorphism.
Hint.
Show that the elements
\begin{equation*}
[g]([h]-1) \text{ for } g \in \{g_1,\dots,g_n\}, h \in H
\end{equation*}
6.
With notation as in Exercise 5, prove that the diagram in Figure 2.3.14 commutes, where \(S\) is given by \(S(x) = x ([g_1] + \cdots + [g_n])\text{.}\)
7.
Prove Theorem 2.3.10.
Hint.
Quotient by the commutator subgroup of \(H\) to reduce to the case where \(H\) is abelian. Apply the classification of finite abelian groups to write \(G/H\) as a product of cyclic groups \(\ZZ/e_1 \ZZ \times \cdots \times \ZZ/e_m \ZZ\text{.}\) Let \(f_i\) be an element of \(G\) lifting a generator of \(\ZZ/e_i \ZZ\) and put \(h_i = f_i^{-e_i} \in H\text{.}\) In the notation of Exercise 5 we have \(0 = \delta(f_i^{e_i} h_i)\text{,}\) which can be rewritten as \(\delta(f_i) \mu_i\) for some \(\mu_i \in \ZZ[G]\) congruent to \(e_i\) modulo \(I_G\text{.}\) Now check that
\begin{equation*}
n \mu_1 \cdots \mu_m \equiv [g_1] + \cdots + [g_n] \pmod{I_H \ZZ[G]}\text{.}
\end{equation*}
For more details, see [38], Theorem VI.7.6.
8.
Let \(L/K\) be an extension of number fields such that \(\Cl(L)\) is trivial. Prove that \(L\) contains the Hilbert class field of \(K\text{.}\) Then deduce the following corollary: the class field tower of \(K\) stabilizes if and only if \(K\) is contained in some number field of class number 1.
Hint.
Take the compositum of \(L\) with the Hilbert class field of \(K\text{.}\)
