CLASS FIELD THEORY. P. Stevenhagen

CLASS FIELD THEORY P. Stevehage Exlicit Algebraic Number Theory Oberwolfach Semiar November 2002

1. Class field theory: ideal grous Class field theory: ideal grous The Kroecker-Weber theorem shows that the slittig behavior of rimes i a abelia extesio L of Q is very simle: it oly deeds o the residue class of modulo the coductor of L. This observatio has a log history goig back to Fermat ad Euler. Classical examles A rime umber is a sum = x 2 + y 2 = (x + iy)(x iy) of two squares if ad oly if it does ot remai rime i the rig of Gaussia itegers Z[i]. This is the rig of itegers of the cyclotomic field Q(ζ 4 ), ad Fermat already kew is a sum of 2 squares if ad oly if it is ot cogruet to 3 mod 4. Euler studied similar roblems, such as the determiatio of the ratioal rimes that occur i the factorizatio of umbers of the form x 2 ay 2 with a Z fixed ad x,y Z ragig over airs of corime itegers. This comes dow to the determiatio of the rimes for which the Legedre symbol ( a ) has a give value, ad the umerical observatio of Euler was that this value oly deeds o mod 4 a. This statemet is essetially equivalet to the quadratic recirocity law. I moder termiology, we would say that the abelia extesio Q( a) of Q is cotaied i the cyclotomic field Q(ζ 4 a ), so the slittig behavior of a rime i Q( a) (i.e. the value of the Legedre symbol ( a ) ) is determied by the slittig behavior of i Q(ζ 4 a ), i.e. by the residue class of mod 4 a. The questio whether a rime is rereseted by the quadratic form X 2 ay 2, i.e., = x 2 ay 2 for certai x,y Z, is already more comlicated, sice this requires that there is a ricial rime ideal i Z[ a] of orm. I Fermat s examle a = 1, the resultig rig Z[i] is a ricial ideal domai, but as soo as this is o loger the case, the situatio is much more difficult. Whe we take a = 5, we are dealig with the rig Z[ 5] that has a class grou of order 2, ad the ratioal rimes that are the orm of a ricial ideal x + y 5 are exactly the rimes that slit comletely i the quadratic extesio Q( 5,i) of Q( 5). As this extesio field is cotaied i the cyclotomic extesio Q(ζ 20 ), the solvability of = x 2 + 5y 2 is equivalet to beig equal to 5 or cogruet to 1 or 9 modulo 20, a result cojectured by Euler i 1744. For other values of a, the situatio is eve more comlicated. For istace, for a = 27 Euler cojectured aroud 1750 that is of the form = x 2 +27y 2 if ad oly if 1 mod 3 ad 2 is a cube modulo. This is a secial case of a more geeral questio suggested by the quadratic recirocity law: do there exist recirocity laws for owers higher tha 2? I order for this questio to be iterestig for geeral > 2, oe restricts to rimes 1 mod, for which the -th owers i F = (Z/Z) have idex i the full grou, ad asks which coditios o the rime esure that some fixed iteger a is a -th ower modulo. This meas that we are lookig for a characterizatio of the ratioal rimes 1 mod that slit comletely i the field Q( a) or, equivaletly, the ratioal rimes that slit comletely i the ormal extesio M = Q(ζ, a). For > 2, this is ot a abelia extesio of Q for most a, ad we will see that this imlies that the slittig behavior of a ratioal rime i M/Q is ot determied by a cogruece coditio o. I fact, fidig a recirocity law goverig the slittig of rimes i o-abelia extesios is a roblem that is still very much oe today. 2 relimiary versio, November 7, 2002

Sectio 1 Goig back to Euler s cojecture for the secial case where = 3 ad a = 2, we see that the ratioal rimes that slit comletely i Q(ζ 3, 3 2) should be the rimes of the form = x 2 + 27y 2. This is ot a cogruece coditio o, but it states that a rime i K = Q(ζ 3 ) of rime orm 3 slits comletely i the abelia extesio K( 3 2)/K if ad oly if it is geerated by a elemet π = x + 3y 3 = (x + 3y) + 6yζ 3. As x ad y do ot have the same arity, this meas that the rime ca be geerated by a elemet π O K = Z[ζ 3 ] that is cogruet to 1 mod 6O K. Geerators are determied u to multilicatio by elemets i O K = ζ 6, so we see that rovig Euler s cojecture o the cubic character of 2 comes dow to showig that a rime of K slits comletely i K( 3 2)/K if ad oly if is a ricial ideal whose geerator is trivial i (O K /6O K ) / ζ 6. This is a cyclic grou of order 3, so we have a abstract isomorhism (1.1) (O K /6O K ) /im[o K] Gal(K( 3 2)/K), ad rimes whose class is the uit elemet should slit comletely. As Arti realized i 1925, this suggests strogly that the isomorhism above mas the class of rime to its Arti symbol, just like the familiar isomorhism (Z/Z) Gal(Q(ζ )/Q) for abelia extesios of Q mas ( mod ) to its Arti symbol. Note that the ramifyig rimes 2 ad (1 ζ 3 ) 3 i K( 3 2)/K are exactly the rimes dividig the coductor 6O K. The tamely ramified rime 2 divides the coductor oce, ad the wildly ramified rime (1 ζ 3 ) divides it twice, a heomeo that is well kow for coductors over Q Towards the mai theorem The two extesios K K(i) for K = Q( 5) ad K( 3 2)/K for K = Q(ζ 3 ) have i commo that they are abelia extesios, ad that the rimes of K that slit comletely i it are the rimes that are ricial ad satisfy a cogruece coditio modulo certai owers of the ramified rimes. I the first case, there are o ramified rimes ad the oly coditio is that be ricial. I the secod case all rimes are ricial, but oly those satisfyig a cogruece modulo 6 slit comletely. A far reachig geeralizatio that oe might hoe to be true would be the followig: for every abelia extesio L/K of umber fields, there exists a O K -ideal f such that all ricial rimes geerated by a elemet π 1 mod f slit comletely i L/K. As divisors of this coductor ideal f oe exects to fid the rimes that ramify i L/K, ad oe ca hoe that, just as for K = Q, the smallest ossible f is divisible exactly by the ramifyig rimes, ad the rimes occurrig with exoet > 1 are the wildly ramifyig rimes. As we have hrased it, the statemet is correct for our two examles, but it fails to hold for K = Q. The reaso is that the slittig rimes i the cyclotomic field Q(ζ ) are the rime ideals Z for which the ositive geerator is cogruet to 1 modulo. A sig chage i the residue class modulo chages the corresodig Arti symbol by a comlex cojugatio, so this eculiar detail is oly relevat to abelia extesios L/Q that are comlex, i.e. extesios i which the real rime is ramified. Whe we take this ito accout, we arrive at the followig weak form of the mai theorem of class field theory. versio November 7, 2002 3

Class field theory: ideal grous 1.2. Mai theorem (weak form). For every abelia extesio of umber fields L/K there exists a O K -ideal f such that all rimes of K that are ricial with totally ositive geerator π 1 mod f slit comletely i L/K. The smallest ideal f oe ca take i 1.2 is the coductor ideal f L/K of the extesio. As we will see, it is exactly divisible by the fiite rimes of K that ramify i L. The wildly ramifyig rimes occur with higher exoet tha 1. For imagiary quadratic fields K, Theorem 1.2 was roved durig the 19-th cetury by Jacobi, Dedekid, Kroecker, Weber ad others. Such K have o real rimes, ad the reaso that their abelia extesios are relatively accessible stems from the fact that they ca be obtaied by adjoiig the values of comlex aalytic fuctios that occur whe oe tries to ivert certai ellitic itegrals. This is somewhat remiiscet of the situatio for Q, where the abelia extesios are obtaied by adjoiig values of the exoetial fuctio e 2πiz at ratioal values of z. For arbitrary umber fields K, work of Hilbert, Furtwägler ad Takagi i the eriod 1895 1919 culmiated i a roof of a result somewhat stroger tha 1.2. I articular, Takagi roved that give K ad f, there exists a maximal abelia extesio H f /K with coductor ideal f; he also gave a exlicit descritio of the corresodig Galois grou Gal(H f /K). For K = Q, we kow that the maximal abelia extesio of coductor is the -th cyclotomic field Q(ζ ), ad that the isomorhism (Z/Z) Gal(Q(ζ )/Q) seds the residue class of a rime to its Arti symbol. I our two examles this was also the case. For K = Q( 5) we had a isomorhism Cl K Gal(K(i)/K) maig the class of a rime to its Arti symbol as the ricial rimes were exactly the rimes that slit comletely i K(i). For K = Q(ζ 3 ) we ca determie the Arti symbol i K( 3 2) for every rime ot dividig 6, ad writig I K (6) for the grou of fractioal O K -ideals relatively rime to 6 we have the Arti ma ψ K( 3 2)/K : I K(6) Gal(K( 3 2)/K) that mas a rime 6 to the Arti symbol (,L/K). Euler s cojecture is that the rimes i the kerel are the rimes geerated by a elemet cogruet to 1 mod 6O K ad Arti s geeralizatio is that the kerel of ψ K( 3 2)/K cosists of all fractioal ideals geerated by a elemet cogruet to 1 (O K /6O K ), so that the Arti ma iduces the abstract isomorhism 1.1. I its full geerality, this is the followig imortat extesio of 1.2 that Arti cojectured i 1925 ad roved 2 years later, usig a clever reductio to the case of cyclotomic extesios due to Čebotarev. 1.3. Arti s recirocity law. For every abelia extesio of umber fields L/K, there exists a O K -ideal f divisible by all fiite rimes that ramify i L such that the Arti ma ψ L/K : I K (f) Gal(L/K) (,L/K) is surjective ad its kerel cotais all ricial ideals geerated by a elemet x O K that is cogruet to 1 mod f ad ositive at the real rimes : K R that ramify i L/K. 4 relimiary versio, November 7, 2002

Sectio 1 Cycles ad ray classes Arti s recirocity law is a very strog statemet that imlies a large umber of relatios betwee the Arti symbols at differet rimes. It suggests that it is coveiet to iclude the ramified real rimes i the coductor f of the extesio, ad to declare a elemet x O K cogruet to 1 mod f if it is cogruet to 1 modulo the ideal art ad ositive at the real rimes i f. The corresodig otio is rovided by the cycles of a umber field. 1.4. Defiitio. A cycle or divisor of a umber field K is a formal roduct f = () with ragig over all rimes of K such that (i) () is a o-egative iteger for all ad () = 0 for almost all ; (ii) () {0,1} if is real ad () = 0 if is comlex. For ay cycle f, the fiite art f 0 = fiite () of a cycle is simly a itegral ideal of the rig of itegers O K of K, while its ifiite art f = ifiite () is a collectio of real rimes of K. As for ideals, we refer to the exoets () as ord (f) ad write f if ord (f) > 0. Divisibility of cycles is defied i the obvious way, so we write f 1 f 2 if ord (f 1 ) ord (f 2 ) for all. Similarly, the greatest commo divisor gcd(f 1,f 2 ) is the cycle with order mi(ord (f 1 ),ord (f 2 )) at. Cogrueces modulo cycles have to be defied i such a way that the quotiet of two itegral elemets x 1,x 2 1 mod f is agai cogruet to 1 mod f, which is ot the case for the usual additive cogrueces. 1.5. Defiitio. Let be a rime of K ad Z 0 a iteger. The a elemet x K is multilicatively cogruet to 1 modulo, otatio x 1mod, if oe of the followig coditios is satisfied. (i) = 0; (ii) is real, = 1 ad x is ositive uder the embeddig : K R ; (iii) is fiite, > 0 ad we have x 1 + A. For a cycle f = () we write x 1mod f if x 1mod () for all. Let I(f) be the grou of fractioal O-ideals a that have ord (a) = 0 for every fiite rime dividig the cycle f. The ricial ideals xo geerated by elemets x 1mod f form a subgrou R(f) I(f) that is sometimes called the ray modulo f. The termiology stems from the fact that we may idetify the ray R( ) i Q with the ositive ratioal half-lie, a ray from the origi. The factor grou Cl f = I(f)/R(f) is the ray class grou modulo f. The ray class grous will aear as the basic abelia Galois grous over K. Examle. For K = Q there is a sigle real rime =, so every cycle of Q is of the form f = () or f = () for some ositive iteger. The corresodig ray class grous are Cl () = (Z/Z) / 1 mod ad Cl () = (Z/Z). I order to describe the structure of geeral ray class grous, we defie the grou (O/f) for a cycle f = f 0 f by (O/f) = (O/f 0 ) f 1. versio November 7, 2002 5

Class field theory: ideal grous Every x K cotaied i the subgrou K(f) K of elemets that are uits at all fiite rimes i f has a residue class i (O/f) cosistig of its residue class i (O/f 0 ) at the fiite comoet ad the sig of (x) at the comoet of a real rime : K R dividig f. 1.6. Proositio. The ray class grou modulo f is fiite ad fits i a exact sequece of fiite abelia grous. 0 (O/f) /im[o ] Cl f Cl 0 Proof. Let P(f) deote the grou of ricial ideals geerated by elemets x K(f). The we have a exact sequece 0 P(f)/R(f) I(f)/R(f) I(f)/P(f) 0 i which the middle term is by defiitio the ray class grou modulo f. The fial term is the ordiary class grou, sice every ideal class i Cl cotais a ideal from I(f) by the aroximatio theorem. The grou P(f) = K(f)/O admits a caoical surjectio to (O/f) /im[o ], ad the kerel cosists by defiitio of the ray R(f) modulo f. This yields the required exact sequece, ad the fiiteess of Cl f follows from the fiiteess of the outer terms. 1.7. Corollary. If a cycle f is divisible by g, the atural ma Cl f Cl g is surjective. Proof. The outer vertical arrows i the diagram 0 (O/f) /im[o ] Cl f Cl 0 ca ca id 0 (O/g) /im[o ] Cl g Cl 0 are obviously surjective, so the same is true for the middle arrow. Ideal grous We wat to characterize the abelia extesios L/K i terms of the kerel of the Arti ma ψ L/K : I(f) Gal(L/K) i 1.3. The roblem is that this kerel deeds o the chose cycle f. If f satisfies the requiremets of 1.3, the so does ay multile of f. The same situatio occurs if we wat to secify a abelia umber field L Q(ζ ) by the subgrou B (Z/Z) to which it corresods. If we relace by a multile m, we obtai aother subgrou B m (Z/mZ) corresodig to L that is equivalet to B i the sese that the atural ma (Z/mZ) (Z/Z) iduces a isomorhism (Z/mZ) /B m (Z/Z) /B. A ideal grou defied modulo f is a grou B(f) satisfyig R(f) B(f) I(f). If f is aother cycle ad B(f ) a ideal grou defied modulo f, we say that B(f) ad B(f ) are equivalet if for every commo multile g of f ad f, the iverse images of B(f) ad B(f ) uder the atural mas I(g) I(f) ad I(g) I(f ) coicide. If this is the case, it follows from 1.7 that we have a isomorhism I(f)/B(f) = I(f )/B(f ) of fiite abelia grous. The otio of equivalece does ot deed o the choice of a commo multile, 6 relimiary versio, November 7, 2002

Sectio 1 ad we obtai a equivalece relatio o the set of ideal grous. The equivalece classes are simly referred to as ideal grous. If a ideal grou B has a reresetative defied modulo f, we deote it by B(f) ad say that B ca be defied modulo f or has modulus f. Before we formulate the mai theorem i its fial form, we still eed to show that the set of moduli of a ideal grou cosists of the multiles of some uique miimal modulus, the coductor of the ideal grou. Over Q, this reflects the fact that a abelia umber field L ca be embedded i Q(ζ m ) if ad oly if m is divisible by the coductor of L. The geeral statemet for ideal grous follows from the followig lemma. 1.8. Lemma. A ideal grou that ca be defied modulo f 1 ad f 2 ca be defied modulo gcd(f 1,f 2 ). Proof. Write f = gcd(f 1,f 2 ) ad g = lcm(f 1,f 2 ) ad H i = B(f i )/R(f i ). By 1.7, all arrows i the commutative diagram φ 1 I(g)/R(g) I(f1 )/R(f 1 ) φ 2 χ 1 I(f 2 )/R(f 2 ) χ 2 I(f)/R(f) are surjective. We ca defie G = φ 1 1 [H 1] = φ 1 2 [H 2] by assumtio, ad we have to show that there exists a subgrou H I(f)/R(f) with iverse image G i I(g)/R(g). The obvious cadidate is H = χ 1 [H 1 ] = χ 2 [H 2 ]. We have χ i φ i [G] = H, so i order to rove that G = (χ i φ i ) 1 [H] we eed to show ker(χ i φ i ) G. From ker φ i = (R(f i ) I(g))/R(g) G we have [(R(f 1 ) I(g)) (R(f 2 ) I(g))]/R(g) G. We claim the equality (R(f 1 ) I(g)) (R(f 2 ) I(g)) = (R(f 1 )R(f 2 )) I(g). The iclusio is the otrivial oe, so let x i O R(f i ) for i = 1,2 be give such that x 1 x 2 O I(g) holds. If is fiite ad divides g, say f 1, it follows from ord (x 1 x 2 ) = 0 ad x 1 1mod f 1 that ord (x 2 ) = 0. Thus x 1 O ad x 2 O are i I(g), which establishes our claim. As we have ker(χ i φ i ) = (R(f) I(g))/R(g), the roof may be cocluded by showig R(f) to be equal to R(f 1 )R(f 2 ). The iclusio R(f) R(f 1 )R(f 2 ) is immediate from R(f) R(f i ) for both i. For x 1mod f the cogrueces y xmod f 1 ad y 1mod f 2 are comatible, so they are satisfied for some y K by the aroximatio theorem. Now the reresetatio xo = xy 1 yo shows that we have xo R(f 1 )R(f 2 ), thereby rovig the other iclusio. The recedig roof is characteristic for may roofs usig ideal grous i the sese that the aroximatio theorem lays a essetial role. I the idèlic formulatio give i the ext sectio the existece of a coductor will be a trivial cosequece of the formalism. If B 1 ad B 2 are ideal grous of K ad f is a commo modulus, we defie their roduct ad itersectio by (B 1 B 2 )(f) = B 1 (f)b 2 (f) ad (B 1 B 2 )(f) = B 1 (f) B 2 (f). We write B 1 B 2 if B 1 (f) B 2 (f) holds. Oe easily checks that all this is ideedet of the choice of the commo modulus f. versio November 7, 2002 7

Class field theory: ideal grous Mai theorem We ca ow formulate the ideal grou versio of the mai theorem of class field theory. 1.9. Mai theorem. Let K be a umber field, Σ K the set of fiite abelia extesios of K cotaied i some fixed algebraic closure ad B the set of ideal grous of K. The there exists a iclusio reversig bijectio Σ K B such that for a extesio L/K corresodig to a ideal grou B with coductor f the followig holds: (1) the rimes dividig the coductor f are the rimes that ramify i L/K, ad the rimes whose square divides f are the rimes that are wildly ramified i L/K; (2) for every multile g of the coductor f, the Arti ma ψ L/K : I(g) Gal(L/K) is a surjective homomorhism with kerel B(g). The ideal grou B corresodig to a abelia extesio L of K determies the Galois grou Gal(L/K) as for every modulus g of B, the Arti ma for L/K iduces a Arti isomorhism (1.10) ψ L/K : I(g)/B(g) Gal(L/K). The slittig behavior of a rime of K i the extesio L is determied by the ideal class of i the geeralized ideal class grou I(g)/B(g). The field L is the uique field corresodig to this ideal grou B ad is kow as the class field of B. This (highly o-trivial) existece of class fields for every give divisio of rime ideals ito classes modulo a cycle accouts for the ame class field theory. It is ossible to give a exlicit descritio of the ideal grou corresodig to a abelia extesio L/K i terms of L. I fact, this descritio follows comletely from fuctorial roerties of the Arti ma. We will list all these roerties i a sigle theorem ad derive them from 1.9. We eed the actio of the orm o ideal grous to formulate it. If f is a cycle i K ad L/K a fiite extesio, we ca view f as a cycle i L by takig f 0 O L as its fiite art ad the roduct of the real extesios of the f as the ifiite art. I this situatio, the ideal orm N L/K : I L I K ca be restricted to yield a orm ma N L/K : I L (f) I K (f) that mas the ray R L (f) i L ito the ray R K (f) i K. I articular, the iverse image of a ideal grou B(f) i K uder the orm yields a ideal grou N 1 L/K B(f) modulo f i L. We deote its equivalece class by N 1 L/K B. 1.11. Theorem. Let K be a umber field, ad L, L 1 ad L 2 fiite abelia extesios of K iside a algebraic closure K with corresodig ideal grous B, B 1 ad B 2. The the followig roerties hold: (1) we have B(g) = N L/K (I L (g)) R(g) for every modulus of B; (2) the ideal grou B 1 B 2 corresods to the comositum L 1 L 2, ad the ideal grou B 1 B 2 corresods to the itersectio L 1 L 2 ; 8 relimiary versio, November 7, 2002

Sectio 1 (3) if L 2 cotais L 1 ad g is a modulus of B 2, the g is a modulus of B 1 ad there is a commutative diagram I(g)/B 2 (g) Gal(L 2 /K) ca res I(g)/B 1 (g) Gal(L 1 /K) relatig the Arti isomorhisms of L 1 ad L 2 over K; (4) if E K is ay fiite extesio of K, the LE E is a fiite abelia extesio corresodig to the ideal grou N 1 E/KB of E. For every modulus g of B there is a commutative diagram I E (g)/n 1 E/K B(g) N E/K I(g)/B(g) Gal(LE/E) res Gal(L/K). Moreover, the ideal grou B 0 corresodig to the abelia extesio L E of K satisfies B 0 (g) = N E/K (I E (g)) B(g); (5) if E K is ay fiite extesio of K, the the ideal grou B E corresodig to the maximal subextesio of E/K that is abelia over K satisfies B E (g) = N E/K (I E (g)) R(g) for each of its moduli g. Proof. Proerty (2) is a geerality o iclusio reversig bijectios that we leave to the reader. For (3), we observe first that the diagram is commutative because of the roerty (,L 2 /K) L1 = (,L 1 /K) of the Arti symbol of the rimes g that geerate I(g). I articular, if R(g) is i the kerel of the Arti ma of the extesio L 2 /K, it is i the kerel of the Arti ma of the extesio L 1 /K. This imlies that g is a modulus for B 1. The commutativity of the diagram i (4) is roved i a similar way. If r is a rime i E lyig above a fiite rime g, it is uramified i LE/E ad oe has (r,le/e) L = (,L/K) f(r/) = (N E/K r,l/k). This also shows that the ray R E (g) is i the kerel of the Arti ma ψ LE/E : I E (g) Gal(LE/E), sice its orm image N E/K (R E (g)) R(g) is i the kerel of ψ L/K. As the restrictio ma o the Galois grous is ijective, we have ker(ψ LE/E ) = N 1 E/KB(g) as the ideal grou corresodig to the extesio LE of E. Usig Galois theory, we see that the cokerels of the vertical mas give a isomorhism I(g)/N E/K (I E (g)) B(g) Gal((L E)/K, ad the restrictio roerty (,L/K) L E = (,(L E)/K) of the Arti symbol shows that this is the Arti isomorhism for the extesio L E of K. It follows that B 0 (g) = N E/K (I E (g)) B(g) is the ideal grou of L E over K. I order to derive the basic statemet (1) from this we take E/K abelia i the revious argumet ad g a modulus of the corresodig ideal grou B E. Settig L equal to the class field of R(g), we have a iclusio E L from B E R(g) ad from what we have just roved we fid B E (g) = N E/K (I E (g)) R(g). versio November 7, 2002 9

Class field theory: ideal grous Fially, for roerty (5), we aly this argumet oce more with E/K fiite, g a modulus of the ideal grou of the maximal subextesio E 0 E that is abelia over K ad L the class field of R(g). This yields L E = E 0 ad the roerty follows. Ray class fields The abelia extesio H f of K corresodig to the ray R(f) modulo a cycle f is kow as the ray class field modulo f. They ca be viewed as geeralizatios of the cyclotomic fields i the sese of Kroecker-Weber to arbitrary K. By the mai theorem, they have the followig roerties. 1.12. Theorem. Let K be a umber field with maximal abelia extesio K ab, f a cycle of K ad H f K ab the ray class field modulo f. The H f is the maximal abelia extesio of K iside K ab i which all rimes of the ray R(f) slit comletely. The extesio H f /K is uramified outside f, ad we have a Arti isomorhism Cl f Gal(Hf /K). The field K ab is the uio of all ray class fields of K iside K ab. Examle. For K = Q the ray class fields ca be give exlicitly as H = Q(ζ + ζ 1 ) ad H = Q(ζ ). I order to rove this, oe alies (4) of 1.11 with E = Q(ζ ) ad L = H. For every rime i Q(ζ ) that does ot divide, the orm N Q(ζ )/Q() = f(/) Z is i the ray R( ), so the left vertical arrow is the zero ma. This imlies that LE = H (ζ ) equals E = Q(ζ ), so H is cotaied i Q(ζ ). As we kow the Galois grou Gal(H /Q) = Cl = (Z/Z) we have H = Q(ζ ) as stated. The real field H H is cotaied i the maximal real subfield Q(ζ + ζ 1 ) of the cyclotomic field, ad it must be equal to it as we have already see that its Galois grou over Q is Cl = (Z/Z) / 1 mod. A ray class field of secial imortace is the ray class field modulo the trivial cycle f = 1 of K. It is kow as the Hilbert class field of K. As the ray class grou modulo the trivial cycle is the ordiary class grou Cl K of K, we have a Arti isomorhism ψ H/K : Cl K Gal(H/K) betwee the class grou of K ad the Galois grou over K of the maximal abelia extesio H of K that is uramified at all rimes of K. Moreover, the rimes that slit comletely i H/K are the ricial rime ideals i the rig of itegers of K. This is a rather surrisig relatio: it is ot at all obvious that the size of a certai uramified extesio of K should be related to the class grou of K, which measures how much the rig of itegers of K differs from a ricial ideal rig. O the other had, this relatio is extremely useful as it eables us to study the class grou of a umber field K by costructig uramified abelia extesios of K. I this cotext, oe also uses a slightly larger field kow as the strict or arrow Hilbert class field. It is the maximal abelia extesio of K that is uramified at all fiite rimes of K. 10 relimiary versio, November 7, 2002

Sectio 2 2. Class field theory: idèles The formulatio of class field theory as give i the recedig sectio is the classical formulatio usig ideal grous. From a comutatioal oit of view, these grous are ofte a coveiet tool as they have a simle defiitio that makes them well-suited for most exlicit comutatios. It is however somewhat aoyig that every roof ivolvig ideal grous starts by the choice of a commo cycle modulo which everythig is defied, ad the ed of the roof is the observatio that the result obtaied is ideedet of the choice of the commo modulus. I order to avoid the choice of moduli, say i the case of base field Q, it is clear that oe should ot work with the grous (Z/Z) for varyig, but ass to the rojective limit Ẑ = lim(z/z) = Z from the begiig ad defie the Arti ma o Ẑ rather tha o a ideal grou I Q () for some large. We see that for the ratioal field, this large grou becomes a roduct of comletios at all fiite rimes of the field. Subgrous of the idèle grou I the geeral case, oe also eeds the real comletios i order to kee track of the sig coditios at the real rimes. Chevalley observed that a very elegat theory results if oe takes the roduct of the uit grous at all comletios of the umber field, i.e. the idèle grou J of K, ad writes all ray class grous as surjective images J Cl f. As the idèle grou J cotais a subgrou (2.1) K = K {1} J for each rime, we obtai a local Arti ma for each comletio K of K. This oit of view eables us to describe the relatio betwee the global abelia extesio L/K ad the local extesios L q /K, thus givig rise to a local class field theory. Moreover, it yields i a atural way a direct descritio of the ower of a rime dividig the coductor of a extesio L/K that stregthes the qualitative descritio of 1.9(1). I order to describe the oe subgrous of the idèle grou J of K, we look at the oe subgrous of the comletios K first. If is a fiite rime, a basis of oe eighborhoods of the uit elemet 1 K cosists of the subgrous U () K defied by { U () U = A if = 0; = 1 + if Z >0. If is real, we have K = R. Every oe subgrou of the multilicative grou R cotais the grou R >0 of ositive real umbers as R >0 is geerated by ay oe eighborhood of 1 R. The oe subgrous of K are therefore U (0) = K ad U (1) = K,>0. versio November 7, 2002 11

Class field theory: idèles Fially, if is comlex, the oly oe subgrou of K is the trivial subgrou U (0) = K, which is geerated by every oe eighborhood of 1 K = C. With this otatio, we have for each cycle f = () of K a subgrou (2.2) W f = U() J. 2.3. Proositio. A subgrou of the idèle grou J of K is oe if ad oly if it cotais W f for some cycle f of K. Proof. As almost all exoets () i (2.2) are equal to zero, the defiitio of the idèle toology shows that W f is a oe subgrou of J. Coversely, if H J is a oe subgrou of J, we must have W f H for some f as every oe eighborhood of 1 J geerates some W f. Ray classes as idèle classes It follows from 2.2 that a subgrou of the idèle class grou C = J/K is oe if ad oly if it cotais the homomorhic image D f of some subgrou W f J. We have a caoical isomorhism J/K W f C/Df for the quotiets of the basic oe subgrous D f C. 2.4. Theorem. For every cycle f of K there are isomorhisms J/K W f C/Df Clf = I(f)/R(f) such that the class of a rime elemet π at a fiite rime f i J/K W f or C/D f corresods to mod R(f) i Cl f. Proof. Write f = (), ad defie a ma φ : J Cl f = I(f)/R(f) (x ) ord (x 1 x ) mod R(f), fiite where x K is a elemet that satisfies x 1 x 1mod () for all rimes dividig f. Such a elemet exists by the aroximatio theorem, ad it is uiquely determied u to multilicatio by a elemet y K satisfyig y 1mod f. By defiitio of R(f), the ma φ is a well defied homomorhism. Its surjectivity is clear as a rime elemet π J at a fiite rime f is maed to mod R(f). It remais to show that kerφ = K W f. Suose we have (x ) ker φ. The there exists x K as above ad y K such that y 1mod f ad ord (x 1 x ) = ord(y). fiite fiite This imlies that x (xy) 1 is a uit at all fiite outside f ad satisfies x (xy) 1 1mod () for f, so we have (x ) xyw f. This roves the iclusio kerφ K W f. The other iclusio is obvious from the defiitio of φ. 12 relimiary versio, November 7, 2002

Sectio 2 2.5. Corollary. Every oe subgrou of C is of fiite idex. Proof. Ay oe subgrou cotais a subgrou D f, which is of fiite idex i C by the fiiteess of the ray class grou Cl f. If B is a ideal grou ad g a modulus for B, we defie the oe subgrou D B C corresodig to B as the kerel D B = ker[c I(g)/B(g)] of the atural ma iduced by 2.3. We have a caoical isomorhism C/D B I(g)/B(g) that mas the class of a rime elemet π at a fiite rime g to ( mod B(g), ad it follows from the defiitio of equivalece of ideal grous that D B deeds o B, but ot o the choice of the modulus g. 2.6. Proositio. The corresodece B D B is a iclusio reservig bijectio betwee the set of ideal grous of K ad the set of oe subgrous of the idèle class grou C. The coductor f of a ideal grou B is the smallest cycle satisfyig D f D B. From the obvious equality D f1 D f2 = D gcd(f1,f 2 ), we obtai as a simle corollary of the formalism a statemet that required a roof i 1.8. 2.7. Corollary. If a ideal grou ca be defied modulo f 1 ad f 2, it ca be defied modulo gcd(f 1,f 2 ). The kerel of the Arti ma Combiig the bijectio betwee oe subgrous of C ad ideal grous i 2.6 with the mai theorem 1.9, we see that every fiite abelia extesio L/K corresods to a oe subgrou D L of C for which there is a Arti isomorhism C/D L Gal(L/K) that mas the residue classes of the rime elemets π mod D L for fiite uramified to the Arti symbol (,L/K). I order to describe the subgrou D L of the idèle class grou corresodig to L, we eed to defie the orm N L/K : C L C K o idèle class grous. We kow (cf. A.2) that there is a adèle orm N L/K : A L A K that is the ordiary field orm N L/K : L K whe restricted to L. It ca be give exlicitly as (2.8) N L/K ((x q ) q ) = ( q N Lq /K (x q )). Here q ad rage over the rimes of L ad K, resectively. The orm mas the uit grou J L = A L ito the uit grou J K ad L ito K, so we have a iduced orm N L/K : C L C K o the idèle class grous. We eed to check that this orm corresods to the orm o ideal class grous uder the isomorhism 2.3. As i the revious sectio, we view a cycle f of K as a cycle i a fiite extesio L whe ecessary, ad use the obvious otatio W L,f J L ad D L,f C L for the corresodig subgrous i J L ad C L. For a cycle f of K we have N L/K [W L,f ] W K,f ad N L/K [D L,f ] D K,f. versio November 7, 2002 13

Class field theory: idèles 2.9. Proositio. Let L/K be a fiite extesio ad f a cycle of K. The there is a commutative diagram C L /D L,f IL (f)/r L (f) N L/K N L/K C K /D K,f i which the horizotal isomorhisms are as i 2.3. IK (f)/r K (f) Proof. The commutativity of the diagram may be verified o rime elemets π q at fiite rimes q of L outside f, sice these classes geerate C L /D L,f. For such rime elemets we have N L/K (π q ) = N Lq /K (π q ) by 2.8, ad by the defiitio of extesio valuatios we have N Lq /K (π q ) A = f(q/). It follows that the diagam commutes. 2.10. Proositio. Let L be a fiite extesio of K. The there exists a cycle f of K such that D K,f is cotaied i N L/K C L ad all rimes dividig f are ramified i L/K. I articular, N L/K C L is oe i C K. Proof. With [L : K] =, we have N L/K J L U for all rimes. As U cotais a oe eighborhood of 1 U, oe has U U (k) for some k Z >0. If q is uramified, the idetity N Lq /K (x + yπ k ) = N Lq /K (x) + Tr Lq /K (y)π k mod k+1 A for x,y A q ad the surjectivity of the orm ad trace ma o the residue class field extesio k k q easily imly that we have N Lq /K [U q ] = U. This roves our roositio, as it imlies N L/K J L W K,f for some f divisible oly by ramifyig rimes. 2.11. Theorem. For ay fiite extesio L/K there exists a cycle f i K that is oly divisible by ramifyig rimes ad a isomorhism C K /N L/K C L I(f)/NL/K I L (f) R(f) that mas the class of π to the class of for fiite uramified. Proof. Take f as i 2.10, the the isomorhism is obtaied by takig cokerels i the diagram of 2.9. Mai theorem We ca ow give the idèlic versio of the mai theorem of class field theory. Note that so far, oe of the roofs i this sectio relied o the mai theorem 1.9 or its corollaries. 2.12. Mai theorem. Let K be a umber field, Σ K the set of fiite abelia extesios of K cotaied i some fixed algebraic closure ad D the set of oe subgrous of the idèle class grou C of K. The there exists a iclusio reversig bijectio Σ K D 14 relimiary versio, November 7, 2002

Sectio 2 such that for a extesio L/K corresodig to the subgrou D of C the followig holds: (1) D = N L/K C L ; (2) there is a global Arti isomorhism ψ L/K : C/D Gal(L/K) such that the image of a comletio K i C is maed oto the decomositio grou G of i Gal(L/K). It iduces a local Arti isomorhism ψ : K /N Lq /K L q G = Gal(L q /K ) Gal(L/K) for the local extesio at. If is fiite, this local isomorhism mas the local uit grou U oto the iertia grou I G ad the class of a rime elemet π at to the coset of the Frobeius automorhism i G. The idèlic mai theorem 2.12 is similar i cotet to 1.9, but it has several advatages over the older formulatio. First of all, it does without the choice of defiig moduli, thus avoidig the cumbersome trasitios betwee equivalet grous. Secodly, it yields a descritio of the cotributio of a rime that shows the local ature of this cotributio. The statemet i (2) is ot a simle corollary of the idetity D = N L/K C L sice it requires the o-trivial idetity (2.13) K (K N L/K J L ) = N Lq /K L q for the itersectio of the subgrou K C with the kerel N L/K C L of the global Arti ma. From (2), we obtai a descritio of the coductor that ca be used to actually comute it. 2.14. Corollary. Let f L/K = () be the coductor of the abelia extesio L/K. If q is a rime of L that exteds, the () is the smallest o-egative iteger for which the iclusio is satisfied. U () N Lq /K U q As a sulemet to 2.12, there are agai the fuctorial diagrams occurrig i 1.11. Both the statemets ad their derivatio from the mai theorem have a immediate traslatio i terms of the idèle class grou, ad we leave them to the reader. Local class field theory The local Arti isomorhism, which occurs as a corollary of the idèlic versio of global class field theory, leads to a class field theory for local umber fields that is iterestig i its ow right. This local theory ca also be develoed ideedetly from the global theory, ad oe may argue that this i certai ways more atural. Our order of resetatio however follows the history of the subject. As we have formulated global class field theory for umber fields oly, ad ot for fuctio fields of dimesio 1 over fiite fields (i.e. extesios of a fiite field of trascedece degree 1), we obtai a local class field theory for local fields i characteristic 0 oly. The theory i characteristic is highly similar, eve though some of the roofs have to be modified for extesios of degree divisible by the characteristic. versio November 7, 2002 15

Class field theory: idèles 2.15. Proositio. Let F be a fiite extesio of Q for some rime umber ad E/F a fiite abelia extesio with grou G. The there is a caoical isomorhism ψ E/F : F /N E/F E G that mas the uit grou of the rig of itegers of F oto the iertia grou I E/F ad a rime elemet oto the Frobeius residue class mod I E/F. Proof. We ca choose umber fields K ad L that are dese i F ad E, resectively, i such a way that L is G-ivariat ad L G = K. This meas that there are rimes q i L ad i K such that F = K ad E = L q, ad G = G. The global Arti ma for L/K ow iduces a local Arti isomorhism ψ E/F with the stated roerties. I order to rove the caoicity of ψ E/F, we have to show that it does ot deed o the choice of the G-ivariat subfield L E. Thus, let L be aother umber field that is dese i E ad stable uder G. Relacig L by LL if ecessary, we may assume that L is cotaied i L. The K = (L ) G cotais K, ad we have F = K = K r for a rime r. The commutative diagram K r C K /N LK /K C LK Gal(LK /K ) id N K /K res K C K /N L/K C L Gal(L/K); derived from 1.11 (4) shows that L /K ad L/K iduce the same Arti isomorhism for the extesio E/F. The descritio of the local Arti isomorhism give by the recedig roositio is somewhat idirect as the ma is iduced by the Arti isomorhism of a dese global extesio. Oly i the case of a uramified extesio E/F the situatio is very trasaret, as i that case both F /N E/F E ad Gal(E/F) have caoical geerators, ad they corresod uder the Arti isomorhism. Oly relatively recetly, i 1985, Neukirch realized that that the local Arti ma i the geeral case is comletely determied by this fact ad the fuctorial roerties of the Arti symbol. We do ot give the argumet here. 2.16. Mai theorem for local umber fields. Let F be a local umber field, Σ F the set of fiite abelia extesios of F cotaied i some fixed algebraic closure ad H the set of oe subgrous of fiite idex of F. The there exists a iclusio reversig bijectio Σ F H such that for a extesio E/F corresodig to the subgrou H of F the followig holds: (1) H = N E/F E ; (2) there is a Arti isomorhism ψ E/F : F /H Gal(E/F) such that, for oarchimedea F, the uit grou U of the valuatio rig of F is maed oto the iertia grou I E/F ad a rime elemet is maed ito the Frobeius coset modulo I E/F. Note that N E/F E F i 2.16 is ideed a oe subgrou of fiite idex, as it cotais F for = [E : F]. We leave it to the reader to formulate the local fuctorial diagrams, which are aalogous to those i 1.11. 16 relimiary versio, November 7, 2002

Sectio 2 The extesio corresodig to a oe subgrou H of fiite idex i F is called the class field of H. I the global case we have class fields corresodig to oe subgrous of the idèle class grou. Aedix: adèles ad idèles A coveiet way to relate a umber field K to its comletios is give by the adèle rig A K of K that was itroduced by Chevalley aroud 1940. This rig is a large extesio rig of K that is costructed from the comletios K of K at all rime divisors of K, both fiite ad ifiite. We kow that the fiite rimes of K corresod to the o-zero rimes of the rig of itegers O K, whereas the ifiite rimes come from embeddigs of K ito the comlex umbers. We write to deote a rime of either kid, ad take A = K if is ifiite. The adèle rig A K of K is defied as A K = K = {(x ) K : x A for almost all }. Iformally, oe ca say that it is the subrig of the full cartesia roduct of all comletios cosistig of vectors that are almost everywhere itegral. It is a examle of a restricted direct roduct. The toology o such a roduct is ot the relative toology, but the toology geerated by the oe sets of the form S O S for some fiite set of rimes S ad O oe i K. This toology makes A K ito a locally comact rig sice all comletios K are locally comact ad the rigs A are comact for all fiite. We have a caoical embeddig K A K alog the diagoal sice the vector (x) for x K is almost everywhere itegral. We usually view this embeddig as a iclusio ad refer to the elemets of K i A K as ricial adèles. For K = Q we fid A A Q = R Q = {(x,(x ) ) : x Z for almost all }. The oe subset U = ( 1/2,1/2) Z of A Q satisfies U Q = {0}, sice a ratioal umber that is -itegral at all rimes is i Z, ad we have Z ( 1/2,1/2) = {0}. It follows that Q is a discrete subrig of A Q. The closure W = [ 1/2,1/2] Z of U is comact i A Q ad satisfies Q + W = A Q. As the atural ma W A Q /Q is a cotiuous surjectio. it follows that its image A Q /Q is a comact additive grou. Aalogous statemets hold for arbitrary umber fields K. They ca be roved by geeralizig the roof for Q, or by usig the followig theorem. If L/K is a fiite extesio of umber fields, we have a caoical embeddig A K A L that seds (x ) to the elemet (y q ) q that has y q = x whe q. versio November 7, 2002 17

Aedix: adèles ad idèles A.1. Theorem. There is a isomorhism of toological rigs A K L A L such that the iduced mas A K = A K 1 A L ad L = 1 L A L are the caoical embeddigs. Proof. For each rime of K, we have a local isomorhism K K L q L q of K -algebras. Takig the roduct over all, we see that there is a isomorhism for the full cartesia roduct of all comletios. I order to show that this isomorhism iduces the required isomorhism for the adèle rigs, we have to show that give a basis ω 1,ω 2,...,ω of L/K, there is a iduced isomorhism i=1 A ω i q A q for almost all rimes of L. This is clear: for almost all rimes it is true that all ω i are -itegral ad that the discrimiat (ω 1,ω 2,...,ω ) is i A, ad for such our basis is a itegral basis of the itegral closure of O K, i L over O K,. The other statemets follow from the corresodig local statemets for K φ = K φ 1 ad L = 1 L A.2. Corollary. The rig A L is a free algebra of rak [L : K] over A K, ad the orm ma N L/K : A L A K iduces the field orm N L/K : L K o the subrig L A L. The adèle rig of K is a locally comact additive grou, so it comes with a traslatio ivariat measure µ kow as the Haar measure o A K. The measure µ is uiquely determied u to a multilicative costat. ad ca be obtaied as a roduct measure of the Haar measures µ o the comletios K. For ifiite rimes the comletio K is isomorhic to R or C, ad µ is the well kow Lebesgue measure. For fiite rimes we ca take for µ the uique traslatio ivariat measure that satisfies µ (A ) = 1 ad µ ( ) = (N) for Z. Here N = N K/Q () Z >0 is the absolute orm of the rime. We defie the ormalized -adic valuatio x of a elemet x K as the effect of the multilicatio ma M x : K K o the Haar measure µ, i.e. µ (xv ) = x µ (V ) for every measurable subset V K. If is fiite, is the -adic valuatio for which a rime elemet at has valuatio N() 1 = (#A /) 1. For a real rime, the ormalized absolute value is the ordiary absolute value o K = R. However, for comlex the ormalized absolute value is the square of the ordiary absolute value. A.3. Product formula. For every o-zero elemet x K, we have x = 1. 18 relimiary versio, November 7, 2002

Sectio 2 Proof. With this ormalizatio, we have fiite x = (#(O/xO)) 1 for every ozero x O by the Chiese remaider theorem ad the idetity x = (#(O /xo )) 1 for each fiite rime. O the other had, the ormalizatio for ifiite rimes yields ifiite x = σ:k C σ(x) = N K/Q(x) = #(O/xO). The roves the theorem for itegral o-zero x, the geeral result follows by multilicativity. The uit grou of the adèle rig A K is the grou J K = K = {(x ) K : x A for almost all } that is kow as the idèle grou of K. For the toology o this grou we do ot take the relative toology comig from the adèle rig, for which iversio x x 1 is ot cotiuous, but the toology geerated by oe sets of the form S O S with S a fiite set of rimes ad O oe i K. This toology is fier tha the relative toology J iherits from A K, ad it makes J K ito a locally comact grou. Uder the diagoal embeddig, the uit grou K of K becomes a subgrou of J K cosistig of the ricial idèles. By the roduct formula, K is a discrete subgrou of J K, so the factor grou C K = J K /K is agai a locally comact grou, the idèle class grou of K. It is ot a comact grou, sice the volume ma A, τ : J R >0 (x ) x is a cotiuous surjective ma that factors via C K by the roduct formula. It is true that the subgrou C 1 K = (kerτ)/k of C K is a comact grou. This follows from the Dirichlet uit theorem ad the fiiteess of the class umber of K. See Reé Schoof s lecture o Arakelov class grous. versio November 7, 2002 19

3. Kummer theory ad class field theory The followig is a theorem from Galois theory. Kummer theory ad class field theory 3.1. Kummer theory. Let 1 be a iteger ad K a field cotaiig a rimitive -th root of uity ζ. The there is a bijectio {K L K ab : Gal(L/K) = 1} {K W K } betwee abelia extesios L of K of exoet dividig ad subgrous W K cotaiig K that seds a extesio L to the subgrou L K ad a subgrou W K to the extesio L = K( W). If L corresods to W, there is a isomorhism Gal(L/K) (W/K ) = Hom(W/K, ζ ). I articular, oe has a equality [L : K] = [W : K ] i this case. If K is a umber field ad L = K( W) a Kummer extesio of exoet, the ramificatio of L/K ca be bouded i terms of W ad. 3.2. Proositio. Let L = K( W) be a Kummer extesio of the umber field K, ad suose that the elemets x 1,x 2,...,x k O K geerate W over K. The a rime of L satisfies: (1) is totally slit i L/K if ad oly if W is cotaied i K ; (2) if is ramified i L/K, the divides or oe of the x i. Proof. The rime slits comletely i the abelia extesio L/K if ad oly if K ( W) equals K, so we have (1). If does ot divide or oe of the x i, the the olyomials X x i are searable over the residue class field at ad their zeroes geerate uramified extesios of K. If F is a local field, we have a descritio of the Galois grou Gal(E/F) of the maximal extesio E of exoet of F by class field theory: F /F Gal(E/F) uder the local Arti ma. If F cotais a rimitive -th root of uity, Kummer theory tells us that we have E = F( F ), ad that Gal(E/F) is the dual of F /F. This yields a airig of F /F with itself. 3.3. Corollary. Let F be a local umber field cotaiig a rimitive -th root of uity. The F /F is its ow dual uder the airig F /F F /F ζ (,β) σ ( β) β. Here σ deotes the Arti symbol of i the extesio F( F )/F. The basic roerties of this -th ower orm residue symbol are the followig. 20 relimiary versio, November 7, 2002

Sectio 3 3.4. Theorem. Let F be a local umber field cotaiig a rimitive -th root of uity ad (, ) : F F ζ the -th orm residue symbol. The the followig statemets hold for,β F. (1) (,β) = 1 if ad oly if is a orm from F( β); (2) (,β) = 1 if F is o-archimedea ad, β ad are i U F ; (3) (, ) = 1 for F ad (,1 ) = 1 for F \ {0,1}; (4) (,β) = (β,) 1. Proof. Proerty (1) follows from the fact that is i the kerel of the Arti ma F Gal(F( β)/f) if ad oly if it is i the orm image of F( β). If F is o-archimedea ad β ad are i U F, the F( β)/f is uramified by 11.5 ad every U F is a orm from F( β). This yields (2). For (3), it suffices to observe that every elemet of the from x β with x F is a orm from F( β) ad substitute the secial values x = 0 ad x = 1. I articular, we have the relatio (β, β) = 1 that yields (, )(, β)(β, )(β, β) = (, β)(β, ) = 1 uo exasio. If a umber field K cotais a rimitive -th root of uity, the same is true for all comletios F = K of K, ad we have a local -th ower orm residue airig (, ), : K K ζ as i 3.3 for each rime. If ad β are i K, the symbol (,β), equals 1 for almost all by roerty (2). The factorizatio of the Arti ma ψ L/K : J K Gal(L/K) for abelia extesios of K via the idèle class grou has the followig global cosequece for the local orm residue symbols. 3.5. Product formula. Suose that K cotais a rimitive -th root of uity ad ad β are i K. The the roduct (,β), over all local -th ower orm residue symbols equals 1. Proof. The restrictio of the global Arti ma ψ : J K Gal(K( β)/k) to K yields the local Arti ma ψ : K Gal(K ( β)/k ), so we have a commutative diagram K J K ψ Gal(K ( β)/k ) (σ ) ψ Gal(K( β)/k) σ. As K is i the kerel of the global Arti ma ψ, we obtai the roduct relatio (,K ( β)/k ) = id K( for the local Arti symbols. The roduct formula follows β) by lookig at their actio o β. We will use the orm residue symbol to discuss the so-called higher recirocity laws. These geeralize ( the famous quadratic recirocity law, which states that the Legedre symbol ) q of two distict odd ratioal rimes ad q satisfies the symmetry roerty ( )( ) q q = ( 1) ( 1)(q 1) 4. versio November 7, 2002 21

Kummer theory ad class field theory This is a rather surrisig result, sice the Legedre symbol ( a ) {±1} of a iteger a Z modulo a rime a is defied by the cogruece ( a ) a ( 1)/2 mod that is i o obvious way symmetric i the argumets a ad. The restrictio to rimes i the quadratic recirocity law is ot at all essetial, sice we ca multily quadratic characters as i 7.* ad defie the Jacobi symbol ( a b) of two corime odd itegers a ad b by the equatio ( a b) = ( a ) ord (b). With this covetio, oe ca formulate the quadratic recirocity law followig Jacobi. 3.6. Quadratic recirocity law. Let a ad b be corime odd itegers, ad write sg(x) {±1} for the sig of a o-zero real umber x. The we have a recirocity ( )( ) a b = ( 1) (a 1)(b 1) 4 + (sg(a) 1)(sg(b) 1) 4. b a Note that the right had side cosists of two factors that are equal to 1 exactly whe a ad b are sufficietly close to 1 i the comletios Q 2 ad Q = R. This heomeo will be exlaied by the geeral ower recirocity law 3.13. Aart from the recirocity law, there are the so-called sulemetary laws ( ) ( ) 1 2 (3.7) = ( 1) (b 1)/2 ad = ( 1) (b2 1)/8 b b for b a odd iteger that eable us to comute ay Jacobi symbol ( a b) without factorig the iteger b ito rimes. For a -th ower recirocity law for arbitrary 2, we eed a character of order o the uit grou of residue class field F that has its image i the grou ζ of -th roots of uity. Such a ma ca oly be defied i a caoical way for all rimes if the base field cotais a rimitive -th root of uity. Thus, we let K be a umber field cotaiig a rimitive -th root of uity ζ. If is a fiite rime of K that does ot divide ad K is a -adic uit, we defie the -th ower residue symbol ( ) ζ by the cogruece (3.8) ( ) (N 1)/ mod. Note that this is well defied as all -th roots of uity are distict modulo ad (N 1)/ is a -th root of uity i the residue class field k = O K /. The ame of the symbol is exlaied by the fact that the kerel k of the character ( ) : k ζ cosists of the residue classes i k that are -th owers. The -th ower residue symbol of K modulo a arbitrary fractioal ideal b of K is defied by ( ) = ( ) ord (b). b / S() Here S() = S () stads for the set of rimes of K that either divide or occur i the factorizatio of (). Note that we do ot require b to be i the grou I() = I () 22 relimiary versio, November 7, 2002

Sectio 3 of fractioals ideals corime to ad, but simly set ( ) = 1 for I(). With this defiitio, the -th ower symbol ( ) b is urestrictedly multilicative i b for fixed, ad the idetity ( ) ( b b ) = ( ) b holds wheever ad are uits at the rimes occurrig i the factorizatio of b. The ower residue symbol for arbitrary,β K is defied as (3.9) ( ) ( ) = = ( ). β (β) S() We ( have already see i the itroductory art of sectio 1 that the ower residue symbol ) for a rime outside S() equals 1 exactly whe slits comletely i the extesio K( )/K, i.e. whe the Arti symbol (,K( )/K) is the idetity. The recise relatio is the followig. 3.10. Proositio. Let K be a umber field cotaiig a rimitive -th root of uity, K a o-zero elemet ad b a fractioal ideal i I(). The we have ( ) = σ b( ) b, with σ b the Arti symbol of b i Gal(K( )/K). Proof By 11.5, the extesio K( )/K is uramified at all rimes i I(), so the right had side is a well-defied root of uity. As both sides are multilicative i b, we ca assume that b = is a rime ideal i I(). For this we have a cogruece σ ( ) ( ) N 1 = (N 1)/ ( ) mod, ad this imlies equality as -th roots of uity that are cogruet modulo are equal. 3.11. Corollary. Let f be the coductor of K( )/K ad 1, 2 rimes i I() that are i the same class i Cl f. The the ower residue symbols ( ) ad ( 1 2 are equal. ) Examle. Take K = Q ad a Z a odd iteger, the the coductor of Q( a)/q has fiite art Q( a) ad is divisible by the ifiite rime of Q exactly whe a is egative. We readily obtai Euler s observatio that ( a ) is always determied by the residue class of modulo 4 a. For a 1 mod 4 it oly deeds o mod a, ad for a > 0 the behavior of the residue classes mod 4a ad mod 4a is the same. I the geeral case, we eed local orm residue symbols to describe the quotiet of the symbols ( β ) ad ( ) β. As our umber field K cotais a rimitive -th root of uity, we have a local -th ower orm residue airigs (, ), : K K ζ for each rime. They bear the followig relatio to the -th ower residue symbol. versio November 7, 2002 23