Proposition 2.2.1.
For \(K = \QQ(\sqrt{-5}), L = \QQ(\sqrt{-5}, \sqrt{-1})\text{,}\) let \(\gothp\) be a prime of \(\gotho_K\text{.}\) Then \(\gothp\) splits in \(L\) if and only if \(\gothp\) is principal.
Section 2.2 Generalized ideal class groups and the Artin reciprocity law
Reference.
Subsection An example (continued)
Before proceeding to generalized ideal class groups, we continue a bit with Example 2.1.1 to illustrate what is about to happen.
Proof.
First suppose \(\gothp = (p)\text{,}\) where \(p \neq 2, 5\) is a rational prime that remains inert (i.e., does not split and is not ramified) in \(K\text{.}\) This happens if and only if \(-5\) is not a square mod \(p\text{.}\) In this case, one of \(-1\) and \(5\) is a square in \(\FF_p\text{,}\) so \(\gotho_K/\gothp\) contains a square root of one of them, hence of both (since \(-5\) already has a square root there). Thus the residue field does not grow when we pass to \(L\text{,}\) that is, \(\gothp\) is split.
Next suppose \(p \neq 2,5\) is a rational prime that splits as \(\gothp \overline{\gothp}\text{.}\) If \(\gothp = (\beta)\) is principal, then the equation \(x^2 + 5y^2 = p\) has a solution in \(\ZZ\) (namely, for \(x + y \sqrt{-5} = \beta\)), but this is only possible if \(p \equiv 1 \pmod{4}\text{.}\) Then \(p\) splits in \(\QQ(\sqrt{-1})\) as well, so \(p\) is totally split in \(L\text{,}\) so \(\gothp\) splits in \(L\text{.}\)
Conversely, suppose \(\gothp\) is not principal. Since there are only two ideal classes in \(\QQ(\sqrt{-5})\text{,}\) we have \(\gothp = \alpha (2, 1 + \sqrt{-5})\) for some \(\alpha \in K\text{.}\) Thus \(\Norm(\gothp) = |\Norm(\alpha)| \Norm(2, 1 + \sqrt{-5})\text{.}\) If \(\alpha = x + y \sqrt{-5}\) for \(x,y \in \QQ\text{,}\) we then have \(p = 2(x^2 + 5y^2)\text{.}\) Considering things mod 4, we see that \(2x\) and \(2y\) must be ratios of two odd integers, and \(p \equiv 3 \pmod{4}\text{.}\) Thus \(p\) does not split in \(L\text{,}\) so \(\gothp\) cannot split in \(L\text{.}\)
The only cases left are \(\gothp = (2, 1+\sqrt{-5})\text{,}\) which does not split (see above), and \(\gothp = (\sqrt{-5})\text{,}\) which does split (since \(-1\) has a square root mod 5).
Remark 2.2.2.
As a bonus, note that in Proposition 2.2.1, for any ideal \(\gotha\) of \(\gotho_K\text{,}\) \(\gotha \gotho_L\) is principal. (To check this, it suffices to verify that \((2, 1+\sqrt{-5})\gotho_L = (1+\sqrt{-1})\gotho_L\text{.}\)) This is a special case of the principal ideal theorem (Theorem 2.3.1).
Subsection Generalized ideal class groups
In this section, we formulate (without proof) the Artin reciprocity law for an arbitrary abelian extension \(L/K\) of number fields. This map will generalize the canonical isomorphism, in the case \(K = \QQ\text{,}\) of \(\Gal(L/\QQ)\) with a subgroup of \((\ZZ/m\ZZ)^*\) for some \(m\text{,}\) as well as the splitting behavior we saw in the previous example. Before proceeding, we need to define the appropriate generalization of \((\ZZ/m\ZZ)^*\) to number fields.
Definition 2.2.3.
Recall that the ideal class group \(\Cl(K)\) of \(K\) is defined as the group \(J_K\) of fractional ideals modulo the subgroup \(P_K\) of principal fractional ideals. Let \(\gothm\) be a formal product of places of \(K\text{;}\) you may regard such a beast as an ordinary integral ideal together with a nonnegative coefficient for each infinite place.
Let \(J_K^\gothm\) be the group of fractional ideals of \(K\) which are coprime to each finite place of \(K\) occurring in \(\gothm\text{.}\) Let \(P_K^\gothm \subseteq J_K^\gothm\) be the group of principal fractional ideals generated by elements \(\alpha \in K\) such that:
-
for \(\gothp^e | \gothm\) finite, \(\alpha \equiv 1 \pmod{\gothp^e}\text{;}\)
-
for every real place \(\tau\) in \(\gothm\text{,}\) \(\tau(\alpha) > 0\text{.}\)
(There is no condition for complex places.) Then the ray class group \(\Cl^\gothm(K)\) is defined as the quotient \(J_K^\gothm / P_K^\gothm\text{.}\) A quotient of a ray class group is called a generalized ideal class group.
Subsection The Artin reciprocity law
We imitate the “reciprocity law” construction we made for \(\QQ(\zeta_m)/\QQ\) (Definition 1.1.7) with a general abelian extension of number fields, but this time with no a priori reason to expect it to give anything useful.
Definition 2.2.4.
Let \(L/K\) be a (finite) abelian extension of number fields. For each prime \(\gothp\) of \(K\) that does not ramify in \(L\text{,}\) let \(\gothq\) be a prime of \(L\) above \(K\text{,}\) and put \(\kappa = \gotho_K/\gothp\) and \(\lambda = \gotho_L/\gothq\text{.}\) Then the residue field extension \(\lambda/\kappa\) is an extension of finite fields, so it has a canonical generator \(\sigma\text{,}\) the Frobenius automorphism, which acts by raising to the \(q\)-th power. (Here \(q = \Norm(\gothp) = \#\kappa\) is the absolute norm of \(\gothp\text{.}\)) Since \(\gothp\) does not ramify, the decomposition group \(G_\gothq\) is isomorphic to \(\Gal(\lambda/\kappa)\text{,}\) so we get a canonical element of \(G_\gothq\text{,}\) called the Frobenius of \(\gothq\text{.}\) In general, replacing \(\gothq\) by \(\gothq^\tau\) for some \(\tau \in \Gal(L/K)\) conjugates both the decomposition group and the Frobenius by \(\tau\text{;}\) since \(L/K\) is abelian in our case, that conjugation has no effect. Thus we may speak of “the Frobenius of \(\gothp\)” without ambiguity.
Now for \(\gothm\) divisible by all primes of \(K\) which ramify in \(L\text{,}\) define the Artin map to be the homomorphism
\begin{equation*}
J_K^\gothm \to \Gal(L/K) \qquad \gothp \mapsto \Frob_\gothp\text{.}
\end{equation*}
Remark 2.2.5.
The fact that we have to avoid the ramified primes will be a bit of a nuisance later. Eventually we’ll get around this using the adelic formulation (Section 6.4).
At this point, the following miracle occurs.
Theorem 2.2.6. Artin reciprocity.
There exists a formal product \(\gothm\) of places of \(K\text{,}\) including all (finite and infinite) places over which \(L\) ramifies, such that \(P_K^\gothm\) belongs to the kernel of the Artin map.
Proof.
We will deduce this much later from a corresponding statement made in the language of adèles and idèles. See Theorem 6.4.3 and Proposition 6.4.9.
In particular, we get a map \(J_K^\gothm/P_K^\gothm \to \Gal(L/K)\) which turns out to be surjective (see Exercise 5), but now we don’t have the Kronecker-Weber theorem to explain this.
Definition 2.2.7.
Define the conductor of \(L/K\) to be the smallest formal product \(\gothm\) for which the conclusion of Theorem 2.2.6 holds. We say \(L/K\) is the ray class field corresponding to the product \(\gothm\) if \(L/K\) has conductor dividing \(\gothm\) and the map \(J_K^\gothm/P_K^\gothm \to \Gal(L/K)\) is an isomorphism.
Theorem 2.2.8. Existence of ray class fields.
Every formal product \(\gothm\) has a ray class field.
Proof.
Again, we will deduce this later from a statement in the adelic language. See Theorem 6.4.4 and Remark 6.4.10.
Example 2.2.9.
The ray class field of \(\QQ\) of conductor \(m\infty\) is \(\QQ(\zeta_m)\text{.}\) The ray class field of \(\QQ\) of conductor \(m\) is the maximal real subfield of \(\QQ(\zeta_m)\text{.}\)
Remark 2.2.10.
Unfortunately, for number fields other than \(\QQ\text{,}\) the abelian extensions are not all generated by roots of unity (see for example Exercise 5), and in general we do not have an explicit description of the ray class fields as being generated by particular algebraic numbers. A salient exception is the imaginary quadratic fields, for which the theory of elliptic curves with complex multiplication provides such numbers. Also, if we were to work with function fields instead of number fields, the theory of Drinfeld modules would do something similar.
This gap in our knowledge, also referred to as Hilbert’s 12th Problem, will make establishing class field theory somewhat more complicated than it would be otherwise. In particular, the proof of Theorem 2.2.8 is rather inexplicit; see Section 7.4. By contrast, in the local setting we do have an explicit construction; see Section 4.6.
Exercises Exercises
1.
For \(\gothp\) a prime ideal of \(K\) and \(L/K\) an abelian extension in which \(\gothp\) does not ramify, let \(\Frob_{L/K}(\gothp) \in \Gal(L/K)\) be the Frobenius of \(\gothp\text{.}\) Prove that Frobenius obeys the following compatibilities:
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If \(M/L\) is another extension with \(M/K\) abelian, \(\gothq\) is a prime of \(L\) over \(\gothp\text{,}\) and \(M/L\) is unramified over \(\gothq\text{,}\) then \(\Frob_{M/K}(\gothp)\) restricted to \(L\) equals \(\Frob_{L/K}(\gothp)\text{.}\)
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In this notation, \(\Frob_{M/L}(\gothq) = \Frob_{M/K}(\gothp)^{f(\gothq/\gothp)}\text{,}\) where \(f\) denotes the residue field degree.
2.
Find a formula for the order of \(\Cl^\gothm(K)\) in terms of the order of \(\Cl(K)\) and other relevant stuff.
3.
Use Exercise 2 to give a formula for the order of \(\Cl^{\gothm}(\QQ(\sqrt{D}))\) for \(D\) odd and squarefree, in terms of the prime factors of \(\gothm\) and \(D\) and the class number of \(\QQ(\sqrt{D})\text{.}\)
4.
Find the ray class field of \(\QQ(i)\) of conductor \((3)\text{,}\) and verify Artin reciprocity explicitly in this case.
5.
Let \(K\) be a quadratic number field. Prove that \(K\) admits an abelian extension which is not contained in \(K(\zeta_n)\) for any positive integer \(n\text{.}\)
6.
Let \(K\) be a number field and let \(\gothm\) be a formal product of places of \(K\text{.}\) Prove that the ray class field of \(K\) of conductor \(\gothm\) is unique.
Hint.
Let \(L_1, L_2\) be two ray class fields of conductor \(\gothm\text{.}\) Then the compositum \(L = L_1 L_2\) is also abelian and so Artin reciprocity applies to it. Use the compatibility of the Artin symbol with field extensions to deduce that the maps \(\Gal(L/K) \to \Gal(L_i/K)\) have the same kernel, and conclude that \(L_1 = L_2\text{.}\)
