foreign courses this week and every course is going to have one lecture during the day so this is lecture one of the first course which is on classical Theory and the afternoon will be exercises um the whole idea is that actually we don't expect you to sort of really absorb everything of these four courses in five days if you haven't seen much about it it's important that you get a feeling what these things are about that you've heard about it and that you sort of know which people to ask where to look and maybe already sort of get your personal experience actually dealing with these things it's going to be all very concrete I mean if you do Galva Theory it's one theorem you can write it down learn it by heart and say now I know Gala Theory it doesn't work that way in mathematics it's a theorem that you need to use in the same way there's going to be four lectures on class with Theory and you can just project the main theorem of class with you and say that's it and now you know what it is personally and I'm not the only one I think the sort of the Hands-On feeling is important so there will be lots of exercises that do with actual computations of course I'm so old almost as old as I read that we know that even without Paris you can still do things just by hand so many exercises can just be dealt with by hand they're not at all computational um some of them actually are and then you may want to use Parry or magma or your favorite brand of computer algebra system to actually sort of get numerical data and that's of course is towards the arithmetic statistics since we're now going to do Theory this first hour but next week you'll see that once you're very can very easily find say class groups or class Fields you may start wondering what these animals look like sort of on average and how they should behave roughly and you come up with Statistics and the main thing about statistics that's going to be a topic next week is that things behave randomly within the borders of the theorems that exists so as the mathematician we have to first sort of find the theories that exist and if you cannot prove anything more than you hope that the whole thing and all the rest is random and also finding out what random means is not at all trivial think of Cohen lenstra what is a random group that's also a kind of mathematics it's more like philosophy and you come up with a model yeah that happens in Natural Sciences and you're going to test it numerically and if it doesn't work there's maybe a theorem that is sort of preventing it from being true and you've probably improved things in the end you want to prove things being a mathematician okay that's where we're going for now we're just going to do in a sort of an easy introduction it's 9 A.M on a Monday morning to class with Theory which has a reputation of being somewhat complicated because it's typically not included if you do algebraic number theories or your first course you will learn about number fields in your second course you will do local fields and details and then beyond that there's class with Theory so people think oh there's not be something really complicated but actually it's almost a century old by now and the theorems are not only rather accessible you can also use them in practice so my idea is of this first lecture to give you an introduction to how you get to class with Theory what it says it will be fairly classical meaning that I will use the old-fashioned language that it was created in a century ago tomorrow we'll make it slightly more modern go for e-dels which is the same language but phrase differently so you should view it as a big difference it makes it slightly more modern on Wednesday there will be a nice half I will tell you about the chromological formulation again that's reformulating the same theorems in a different way and then we will apply them in the last two lectures to do the classical reciprocity laws and do something modern on Friday is the idea which is called radio reciprocity which has been proved rather recently finally proved I should say and also has been used recently in predicting for improving go and Landslide conjectures for two class groups of quadratic Fields so there's the plan and let me now start with the easy introduction so algebraic number theory is a prerequisite and if you remember people tell you why would you need algebraic number Theory that's because in number Theory we typically solve equations say in integers or in rational numbers so if you pick your favorite equation and I think many number theories that have to mention their favorite or the world's favorite equation that's going to be this one and that's an equation that you're supposed to solve over the integer so that's the fermar equation you're interested in Solutions over Z and for my header conjecture theorem which took a long phrase in Latin in those days actually writing this down is really something relatively modern Euler could do that could do this but not Pharma he wrote Latin and just had phrases about nth Powers writing it like this is extremely efficient and that's something that we owe to people like eulers that's a century later in the 17th century in the 18th century people um also discovered that there were complex numbers and in the 19th century so what mixed it up I think 17th from my 18th Euler so 19th century where now that's pretty kumbar is sort of the beginning of what you should call Standard algebraic number Theory and the observation is that you may want to do this over the integers but if you allow yourself complex numbers then you can actually Factor it Z to the N minus X3 and there's a product of linear factors and what you need to do that there's a z here I guess and the X there it's a product of n linear factors maybe we want to call this m let me try and stick to M for this one sorry and this is over the complex now what you need is a primitive and through the unity so if you know the exponential function is over the complex numbers you can Define this complex number e to the 2 pi I over M that's a primitive M3 of unity and then your polynomial expression these are algebraic equations we're solving diaphragm equations becomes something that is a product of linear factors but you do need something outside the rational numbers and if you go for complex numbers you're doing geometry so in modern terminology whether you solve equations or go for rational points over whatever base field on your curve that's just phrasing the same statement in more or less fashionable ways and Kumar discovered that if you want to do this you don't need what I was about to say this is over the complex numbers but these elements actually are not just random complex number they live in a numbering in this case it's going to be a cyclotomic ring so this is our first example of a numbering and then what you need is a theory that does for this numbering what uh ordinary arithmetic for Z that you have known since Euclid more or less so here you have primes you can you can look at factorization all these sort of well-known things the question is do you also have them for these rings in this case cyclotomic Rings that's uh Kumar came up with so the algebraic number Theory a and t is essentially doing arithmetic like in z but now we have to develop them for number rings and number rings are things that look like the cytotomic ring and there's a whole Theory so that is your first course in algebraic number Theory and these number Rings live inside things that are called number of fields so there's the field Q Zeta m the nth cycle Atomic field and in general what does algebraic number Theory tell you [Applause] rather than working in the rationals and then the visibility is phrased in terms of integers so you need a ring rather than a field to talk about divisibility you're going to extend it to find an extension of the rationals called number Fields so and they always look like Q Alpha so you can always get them by enjoying something like Zeta M so typically I will have a polynomial f which is reduced for polynomial of alpha over the rationals and just by replacing it by multiple I always take it to demonic with integral coefficients monarch irreducible so that's the way you get a number field and that's not because you like number Fields that's just because the equation at hand forces you to work in a ring which is bigger than Z but not that much bigger so these rings in in here you will have a ring z alpha this is a an extension of degree n where n is the degree of this irreducible polynomial and z alpha will be a module which is free of rank n over Z and you try and do arithmetic in z alpha as you do it in z and what are the theories say so this is what is most proved by Kumar and his successors so the main points are first of all it does depend which Alpha you take this in a way is the right ring if you're working inside the cytomic field the reason being that rather than z alpha you have to find a ring which is called The Ring of integers so we so find the ring okay which is the ring of integers in your number field [Applause] and it can be easy as it is for the cytotomic fields or it can be much more complicated for these Rings these rings of integers yeah this will be free a finite rank and over Z so it looks pretty much like a cyclotomic ring or like a ring z alpha and it of course it also has a definition it's the integral closure of Z in your number field so this is the number field is called K the Ring of energy is called ok so you need to find that ring and that's the ideal ring to do arithmetic in finding it of course may be complicated but we'll get that in a moment and what are the main statements of algebraic number three for these rings so okay there's first a problem but also a solution which is if you want to do arithmetic which is by divisibility and factoring you need primes so the word Primes in Z they are just prime numbers and they're typically taken to be positive for this ring of integers you go for Prime ideals and you still call them Primes and then the statement is that okay it may not be a unique factorization domain but it has at least unique Prime ideal factorization meaning that every non-zero ideal in that ring can be factored uniquely up to ordering as a product of prime ideals so it pretty much looks like the ordinary factorization the only thing is that you have to deal with ideals and when you deal with ideals there is first of all the question is a really different from elements since if these ideals are principled at least they have a generator which is unique up to units multiplication by units and the answer is it's almost correct namely there is a finite abelian group the class group which you also need to define the classical of K or of ok whatever you want which is gotten by taking all the ideals so that's the non-zero okay ideals you take them up to equivalent that is up to scaling by elements so you can phrase it in two ways everything inside okay or you make it a group first you go for fractional ideals and you're mod out by the so-called principle ideals so fractional okay ideals module principle just that do have a generator and this is an abelian group and the first important statement is that this is actually a finite group which means that you don't have what you would have liked just a unique factorization domain but you're not that far away it's called the delicate domain I mean all the big names sort of sort of past and algebraic number Theory Coomer dedicate chromacker Weber you're going to see them all on the Blackboard or at least you can hear them today and secondly as I said if adios are principal they have a generator which is unique up to units so also the question is what is the unit group and there is also a theorem which is called the deviously unit theorem and the original show that this group is finally generated meaning that it has a finite torsion part that's called mu K these are the roots of unity that's actually finite a finite cyclic group times the number of free generators and if you want to specify that number typically people look at embeddings of your number field in the complex numbers just like the cyclotomic ring is a sub ring of the complex numbers but actually there will be more embedding so you look at all of them and some of them actually land up in the reels that's called the real embedding or a real Prime and if it's in C but not in R it's called complex and then they come in pairs complex conjugate pairs obviously and if you count them there will be n embeddings with n the field degree and if you call the number of complex embeddings s and the reals R then you see that R plus these are pairs R plus two s equals the degree m K Over Q these R and S that the signature of the number field So that's its Archimedean character if you want determines the structure of the unit group since the free rank of the unit group will be R plus s minus one these are all very old and classical theorems going back to the 19th century and when people like Enrico and started creating programs like Paris or systems actually finding these objects was something that people especially older people did by hand in those days and some other people never did it at all they were happy with the theory and said okay that's the theory now I know what these things look like but in general it's not so easy namely if someone throws a paranormal at you because there is some number field that comes up in whatever you're doing then finding this ring of integers already is a step which is not completely trivial since what do you do if you're confronted with some Monarch irreducible polynomial and the associate number field K well you just hope that z alpha is anywhere close to the Ring of integers and you need to do some sort of linear algebra over Z you define things like discriminants of such a an order and of the polynomial and you discover that the whole thing is something finite if you look at the discriminant of your polynomial so Delta f is the discriminant of my polynomial f and I assume that you know what a discriminant is of a polynomial there's going to be an integer since my polynomial this morning ZX and that discriminant tells you later a lot about the number field in particular it sort of tells you uh the difference also in a way between Z elf and the Ring of integers This falls apart as the index since the Ring of integers will contain z alpha since the ring available is the maximal order so it will contain any ring z alpha if this Alpha has a molecule reducible polynomial this index is going to be finite and actually it's factors like this and the second Delta is again called to discriminant but that's the intrinsic discriminant of the number field so this is discriminant of f and this discriminant of K it's one of the main invariants of a number of field it actually tells you which primes are going to be ramified so the size of a number field is determined by its degree and it's discriminant and then typically what you need to do is actually figure out which primes occur in this uh in this index which is okay in small examples but if you happen to do something like the number of field sieve you may have an F which has a discriminant which is way bigger than the number you're trying to factor so if Delta F has a thousand digits step one factoring it is not a good idea which means that now in practice people often use the ring z alpha if need be rather than okay which means that you have to sort of review all these theorems and see what remains if you do it for z alpha and the general feeling is that there's only fairly many primes yeah so locally it's almost always correct there's a few bad primes and if you desingularize there you get the full thing but you may not need to be singularize if the point which is singular is sort of far away you never encounter it you simply don't care so the alpha for practical purposes you may enlarge a little bit and then it will behave as the Ring of integers and then Computing the class group is also something which is from the theory it's a minkowsky type argument that is going to be finite it will just tell you that in the class group every ideal class in the class group contains an ideal of an integral ideal of norm no more than the so-called minkowsky constant of the number field and the mikovski constant of a number field is some multiple of the square root of the discriminant so that is something explicit times the square root of the discriminant which means that to find the class group you just look at all the Primes up to the minkowski bound you factor them and you find all relations so we generate relations you find the class group and at the same time you find the unit group so Computing the class group and the unit group is just one in the same algorithm and it's somewhat complicated so this is something that people have seen I don't know it looks like something like 4 over pi to the S it's just completely explicit right and factorial entry n square or something so this is something that is well it's an algorithm you can now think of yourself doing it is not so uh maybe not so easy and at the end of the day you get an answer which is the class group so in arithmetic statistics this is one of the main things you can ask similar questions for us a elliptic curves or building varieties what do you expect to get say for the model a rank in this case it's called coal lenstra will be more about that next week I guess so you have something that you can now compute thanks partially to all these computer systems so there's massive data so class groups a number of field tables now millions of them in the lmfdb which is the L functions modular forms database thank you Samuel uh you'll get to get in touch with I guess during the exercises there's massive data and we can sort of wonder what do class groups look like but at least you should know in principle how to compute them and you do it a few times by hand and then you trust that people oh sorry um that people who create these databases know what they're doing but still sort of knowing which questions you can add to a system and which you can do yourself and what is feasible what is not that's important knowledge for any number of theories these days okay so that is algebraic number Theory and let me get back to where we need to go namely cyclotomic uh Rings since in our particular example there was a cyclotomic ring and cyclotomic rings psychotomic so means gotten by [Applause] the root of unity are in many ways easy as I said this is the general purpose algorithm for z alpha what you need to do first of all you don't need to think what G alpha or the Ring of integers is namely if you take the nth strike Atomic field Q Zeta m then his ring of integers is in fact equal to Z Zeta and the polynomial F that gives rise to it is the nth cyclotomic polynomial so step one already finding is easy and of course then the general theory sort of applies so typically these are totally complex field they don't have real embeddings if m is at least three and in principle you can compute all these things and the second thing that also gives you a big handle on everything around is that you have a group of amorphisms there was a group acting which is zmod MZ star the invertible residue classes modulo m and why is that well here I chose a specific complex numbers in terms of the exponential function if you're a little bit more algebraic then you know that the polynomial F involved which is the nthomic polynomial actually factors as Zeta m to the I where I arranged over Z mod MC star so the roots of the introducible polynomial are just powers of a given primitive amplitude of unity and algebraically of course these are indistinguishable yeah they're just the same thing meaning that you have automorphisms namely if you have a residue Class A it will correspond to nomorphism Phi a and what it does is that it sends data M it raises it to the eighth power which is well defined for any a co-prime to n m and it only matters modulo m and in fact if you do Galva Theory it's not a great 19th century invention this actually becomes the Galva group of the nth cycle Atomic field over the rations and having this group action by an abelian group is a major asset it allows you to look at all the objects that you encounter like the Ring of integers its unit group the class group the ideal group The principle of the old there's an action of the group all over the place and that means that rather than talking about groups yeah group is the same thing as a z module you want to use fancy terminology and what happens is that if you have a Galway action and in particular by an abelia group these things become so this group has a name then whatever the residual class is modulo M or M would be E's notation you get modules over the group ring zrm and if the group is a billion as it is in this case you typically look at the characters and you can split all the objects that you encounter into eigenspaces if you're lucky and the structure is much more it's much easier to holding your hands than at this for arbitrary number fields so in this case the theory of cytomic fields is much more explicit and yeah I should say more accessible than for General number Fields even Galva fields and that's also very visible if you go for the splitting of primes yeah what you typically need to do if you want to do arithmetic in such a big ring here you know what the primes are well more or less right there's a prime number theorem and you sort of feel that you know what ordinary primes are but the prime ideals of such a ring there's more complicated typically you start with a rational Prime so we're up there and then the question is yeah so splitting of primes is a key thing that you need to know meaning that in your number field if it's a general thing Q Alpha as I said you look at this ring z alpha you hope it's the Ring of integers and given P you want to know which primes lie over p and the main theorem there again 19th century is that if you stay away from the singular primes the ones that divide the index all the way up there yeah so for primes that do not divide this index and even if you may not know what the Ring of integers is but at least for any Prime you can check whether it devices agreement of your polynomial and if it doesn't then you certainly know it's not in the index and if that's the case then splitting a prime is essentially factoring the defining polynomial modulo P so P will Factor as a product of primes to certain exponents and the question is can you find the primes lying over P explicitly the answer is yes for the non-singular primes this factorization comes immediately from the factorization of F Bar so it's f modulo p if you factor that over the finite field so let us say GI to the e i is one these are monarch irreducibles but now in fpx then the theorem which is called uh cumulated I think so um tells you that the primes lying over P just come from the polynomials that divide F in fpx namely explicitly the correspondence is as suggested and Pi is just generated by P and GI Alpha so factoring is something that depends on it's just factoring the defining polynomial mod P essentially apart from finally many Primes that is completely General and if you're dealing with a Galva number of fields as we are for the cytomic fields life is a little bit easier in the sense that the outcome is very regular and what is I mean that once you have one prime lying over P you sort of have them all since they're all going to be isomorphic namely if you have your Prime p and you take the extension Primes if there is a group the galba group of K Over Q adding on everything then this action of G is going to be transitive meaning that all the Primes over PR conjugate isomorphic and they're all have the same behavior so then typically there's this famous formula that you learn that you can always count the number of primes in this set it can be anything from from one up to the field degree and you have e f g and these F's and G's have absolutely nothing to do with the f and g that you find there there are canonical letters that are traditional this is called the ramification index that the only letter which is the same as over there so all the Primes dividing P if they occur with a higher exponent they all have the same exponent e f is called the residue field degree namely every Prime has a size meaning that if you look at OK modulo the prime P that's the residue class field KP that will contain FP obviously and it's a finite field extension in the degree is by definition F so this is again F and it has nothing to do with the polynomial that I called fp4 and then G is just the quotient this is the number of primes that actually do divide p this is generalities for arbitrary Galva number fields another important issue is that if you look at the action of G on the primes extending P then you have the stabilizer inside G decomposition group in that subgroup actually hasn't yeah you um this is actually the same thing as the gelber group of the residue class field extension KP over FP and there's an extensional finite field which is generated by frubinus raising to the power P fraught p x goes to x to the p that's the way things work for finite fields and as I said in gold accents you can just raise this Prime P at least if the prime p is unremified so this in general it's a suggestion let me put it like this and it's an isomorphism if p over p is unremified meaning that the exponent is equal to one okay so that means that you always have frobenius elements and if we now get to the cyclotomic fields it's even more explicit so now we take the number of fields equal to km then cyclotomic field and you discover that in this case there's the Galva Group G this is Z mod m z star and that means that inside G there's always the stabilizer of the Prime p and in this case raised to the power p is just a residue class P modem so for primes not dividing m you know exactly what the group is and you also know of course that in fact just you see that if you have FP Zeta M this final field extension over FP the degree is just the order of p in Z mod MC star so for the exercise let me just write down quickly since you don't have a lot of time but this is something that should be your bread and butter cyclotomic Fields they're sort of the prime example of the class Fields we're going to work with so if you take your favorite example of M and I think my favorite example today is m equal 20. then how do you analyze a cyclotomic field gamma theoretically so Q data 20. first of all you decompose M into Prime Powers it's going to be a compositum of Q data four which people know as q i and Q Zeta five so the degree is going to be eight two times four and this is a cyclic extension of degree 4. remote 5z Star right so I'm actually the group zmo 20 Z star but the challenge remainder theorem is a product of two groups corresponding to this being a composition of two cyclotomic fields of conductor four and five and to make this completely explicit you have this Galway Theory and here it's easy since you know that there's a quadratic field ramifying only at five which is real Q squared five and then you just take composites q i square root 5 q square root minus five this is a standard V4 for instance and then you think it's my diagram complete and one of the exercises this afternoon is also why isn't isn't this actually isn't this diagram a hassa diagram why isn't it Complete because for psychotomic Fields more generally that's one of the exercises such a diagram there's the field diagram and there is the group diagram right Galva Theory tells you the subgroups of zemo20z star correspond to subfields of Q Zeta 20. so exercise you find all these things 11 nine minus three something like this and such a diagram you can always put it upside down and it it should be symmetric so there is something lacking here it has to look like this and in fact that the group minus one with the real cycle the reason that I'm writing this actually down explicitly is that it's essentially one of the exercises you have to do something similar and stare at these various extensions these are all subfields of cyclotomic fields so there's group action that you should understand completely and in particular the next thing they're going to talk about is the splitting of primes and that will be essential for athletic statistics namely you can wander given your Primes what frobenius will they have and just to get an idea for Z modem Z star which we now identified as the gelma group of the cyclotomic field over Q we know that P mod m corresponds to the frubinis automorphism of p and I should have said that this rubinius in general depends on the prime over P it's a stabilizer of a prime over p and this is a transitive group action if you change the prime P if you take a different Prime over P you get a conjugate subgroup so in general such a decomposition group is only determined by the prime P at the bottom up to conjugation but we're dealing with a billion extensions which is actually the whole uh key sort of to classical Theory a bean extensions are so much easier it only depends on the prime at the bottom so this p is really an element and it has a fancy name in general it calls an art in symbol because I can realize that this frobenius is Association is the essential thing the forbidius element is somewhere in the galba group and D richly already and that is the beginning of Statistics if you want show that there is equity distribution if you throw away the finely many primes that do ramify that are not co-prime to M then you're left with a set that you can sort of divided over the various buckets and the equity distribution means that each class has the right density so that means that every element corresponds to a set of prime speed which is then congruent to a modem which has the expected density of density one over the order of the group which is the Euler V of n and density well there's two definitions the one that actually is in the proof of dirichlet is called the dirichlet density let me skip the definition for now there's also a density you would Define yourself which is the natural density which is a somewhat stronger concept um either way they usually prove that every residue class gets its share of crimes exactly as you think and that is a key property once you can relate the property of your Prime to splitting behavior and splitting Behavior means frobenius element in the galba group then you immediately get Equity distribution in this case it's very classical it's the theorem of dirichlet on primes and arithmetic progressions but it's valid in a much wider context so starting with dirichlet that is in the 1840s I think that the primes are equidistibuted over the residue classes modulo m you can look at the same question over frobenius elements in Galva groups and the final result is the tubatory of density theorem which is a key theorem that we won't prove that we'll use all the time if so that's the theorem which says if you're number field is galwa actually let me just do it in immediately for L over k for an extension of number of fields rather than over the rationals if the number of field is gallowash you have a gal of extension uh K Over L is now Galway that is now the setting I just replace Q by K then you can always wonder for a given Prime P of your ground field what is the rubinis element and as I said it's only unique up to conjugation in general it's a generator of this stabilizer and that means that you can only ask for this up to conjugation and that means if you take a subset of the Galway group stable under conjugation and you look at the set of primes P of K with the property that it's robinious lens in C that is well defined because of the condition and well it has density exactly what you think How likely is it that a certain Floridians is going to be equal to an element in C it's just the fraction inside the Galva group I just say density as I said there's two versions there is the do you use the density there is the natural density is true for both the natural density is much harder to prove but it's always true and it gives you if you want a first handle on how primes behave sort of statistically in a number of field extensions frobenius elements are random in the sense that they Equity distribute distribute over the Galva group and that's a key theorem so that has been in the making from the 1840s and approved by territory office from 1924. and just like the deviously theorem the proof is analytic we're not doing much proofs here but what you need to do to prove this the way it usually did it is that you define functions called L functions using the characters on the Galva group as I said if you have an a billion Galva group you have characters on them and you can do that to decompose your whatever you're looking at and these L Series complex analytic functions they provide a factorization of the data function of your cyclotomic field and the analytic means yeah so this is the thing that have been pioneered by Riemann the data function the L Series you discover that you get an equity distribution result and the density that you find that comes from the proof is exactly the reason that you have this funny diversary density it's more like residues of complex analytic functions and it's not the natural density so this is a key result that is extremely important and many of the it had been used already I mean this is the final result that is for arbitrary Galva Fields but frobenius for instance already yeah there's various corollaries except you just mentioned one Corollary if you look at if you have any Galway extension of number fields and you look at the primes p that splits completely in the extension [Applause] then you see that's a special case where C is just a unit element of the Galway group since splitting completely means that the frobenius element is the unit element straight extrivially this set has a density one over the field degree [Applause] so for instance yeah and it's these results have been extremely classical already if you take a cyclotomic field you get exactly for instance the equation of dirichlet it's also primes one modem that's split completely in the cyclotomic field have a density one over Phi M also if all the Primes one mod M split completely in your yeah so if the corollary think about it if all or almost all primes one or them split in the extension L over Q then the reason is that L actually is contained in the M cycle Atomic field and let me now finally don't get ready for the key statement on psychotopic fields since we know how to deal with cyclotomic Fields the splitting is easy it just depends on residue place as modulo M and now we use the this water machine even though I'm from Holland I'm used to this this is so fancy this is German style you only find I think it's serum the only place in France okay yeah a little bit too technical this is a statement that we now need that is the theorem which is also 19th century even though the proof is by Hilbert so the theorem was discovered by cronacker and there was a that's the middle of the 19th century Weber gave a proof in the 80s which was still not complete and Hilbert I think used his this theory of decomposition groups and it says that over the rationals cytomic Fields have a billion Galva groups but the converse is also true every extension is cyclotomic and by cyclotomic I mean that L can be viewed as a subfield of Q data M for some m so it's exactly the same thing we can if you know cytotomic extensions yeah then you know in fact all a being extension so a billion number of fields having an ability group are extremely accessible because there are subfields of cytomic fields and I don't have time to see anything on the proof there's many proofs the one by Hubert is sort of classical uh Galva theory of number Fields you can also do it locally yeah tomorrow we're going to talk about local Fields not today and the key step is always in the end you modify your L with roots of unity all kinds of twisting and in the end you create a field over the rationals which is going to be unremified at all Primes and then reaches famous minkowski type result if yeah so this the proof the key step if is totally unrentified meaning that no Prime ramifies then in fact Q is equal to l there's always ramification of the rations and that is the key step in understanding why all the extensions are in fact cyclotomic that's not much of a proof but it's something you should realize and this is exactly which is no longer true over arbitrary number fields which is the direction we're heading right now since what is the main theorem of classical Theory let me just put it here class for Theory it is a theorem that replaces Q by an arbitrary number of it so and it says that every so you fix but just K is a number field rather than q and the statement is that every extension which is a billion or always finite we're dealing with number of fields so obedient means finer the billion and running being contained in the cycloatomic field it will be contained as contained [Applause] in a field that is typically called the Ray class field modulo m let me just call it that for now every being extension is not cyclotomic it is a class field and what does it mean to be a class field it is contained [Applause] in a ray glass field which does the word that's going to replace psychotomic field modulo a modulus app of course there are certain words that need to be defined now what is array class field and what is a modulus but the key statement so here you're saying that there is a group an easy easily understood group Z mod m z star which is the Galva group of Q Zeta M over q and you can understand your field L which is inside it's a quotient of Z mod m z star and the key statement is that a modem well let me put P modem Maybe will correspond to the thrubinius element of P so you understand the Galva group and the corresponding splitting of all the Primes yes and understanding the splitting of prime is knowing what frobenius elements are that is not something which is random it is governed by congruences only knowing what it is more than tells you what splitting behavior is and the reason is that actually L is contained in the corresponding cytotheratomic field and the same statement holds here there is this easily understood group that I will Define in a moment namely the ray class group modulo m it will subject to the Galva group of L over k and the residue class of a prime p at least if P does not divide m just like P mod M I'm assuming here as well that P does not divide m ramifying primes are a bit more complicated and we'll deal with them tomorrow if we do edels right now the class of an unremified Prime in the class group determines its frobenius [Applause] and this map is called the art and map so associating a prime to its rubenius outside of finite set those dividing the modulus m [Applause] this art in map tells you exactly how to understand the earth vector splitting behavior of the primes in the beginning extensions so being extensions are class fields and that means that the splitting behavior only depends on the class of the Prime modulo whatever now there are a few technicalities and that's exactly why some people think that classic Theory looks complicated but if you see it more often then you have five days to look at it and hopefully by the end of the day you should think oh these Ray glass roof they pretty much look like Z modems he Stark let me give you the definition and see that indeed they do look a little bit like zemod MZ star so what is the ray class field modulo M and what is m since since the speed the problem is that in the original version for the rational number field you could just look at the generator of the Prime you take the positive generator yeah if you replace P by minus p which generates the same ideal you get a different for genius element so there's a little technicality you have to deal with units and also as you see in principle this might not be a principle ideal if you're here so what the heck does this mean the residue class of p modulo m and the group you need to Define this rate class group you take all the ideals Co Prime to m and you're mod out by principal ideals that are Let Me Maybe the ray modulo n what are these things first of all your n as I said is not exactly an integer this is an integer and that means it corresponds to a non-zero ideal of Z at least up to sine this m will just be an ideal of the Ring of integers M naught this is a non-zero okay ideal and you have something that's called the infinite part and if you're dealing with infinite primes you've encountered them if you did algebraic number Theory this is not actually an ideal this is just a collection collection of real primes and real primes are just imbalance of K in the real numbers there's only finally many there were R of them up there and you can include them sort of with extra data with your ideal so it's ideal with sign conditions that's called the modulus by this I just mean I take all the invertible ideals Co Prime to m meaning that no primes in this m naught actually occur invertible okay ideals Co Prime to m that makes sense since I only want to Define this for primes that are not dividing M so find the many primes have to be thrown out to get a sensible RTM map and the key statement is that you can of course Define this art in map on the group I am just because it's a free group generated by primes Co Prime to M just sending them to their fubinis here that's an easy map so this IM has this quotient going from here to here is just trivial the key statement is that it does Factor via this class group that the fields is in fact a class field and that means that the kernel contains RM and what does RM this is the group of principal ideals principal okay ideals so they look like Alpha times okay Alpha has any non-zero element and this Alpha is congruent to one modulo this modulus and this modulo well some people give it a star to indicate that there's a little bit non-standard this modulus what does it mean all right it doesn't work it means that just the definition it depends on the primes dividing M and if you have a prime that divides the infinite part that's just an embedding Sigma K and r it means that your Alpha is going to be positive under that embedding so as I said it's your sign conditions Alpha may be required to be positive at certain infinite embeddings real embeddings and these are the primes in the infinite part and for primes are divided the finite part m0 it simply means what you think if you look at the valuation of alpha minus one it is at least the valuation the number of primes number of times your Prime occurs in the factorization of m0 so it's conditioned at finally many primes those dividing the modulus and at finite parentheses what you think it should multiply multiplicatively saying that it's close to one namely disclose VP is the valuation of P just count factors p and it comes with sign conditions so that may look a little bit let me see no wrong so extremely complicated yes foreign class feel sorry Ray Glass groups look like it's easy to check and if you go back to okay go to the rationals then a modulus is just an integer up to sine an ideal MZ maybe with this sign condition at the real embedding so you see that if you take your m equal to m z then the class group that you get is not Z modem Z star but you get modulo plus or minus one the reason being that the units enter here because the generator is only um unique up to sign and if you take so this is an exercise I'm gonna spend time on the towel if you take m and d infinite Prime so whatever infinity and infinity is the embedding of the rationals in reals then this Ray Cloud group is z modem Z star and in general you see that what I did this subgroup I am sorry in I am your scores you can just take all the principal ideals Co Prime to m without imposing the one within condition and in there there is a so-called array modulo M this is the people called the Ray some discussion why it's called Ray let's not go into it right now and you see that this Ray class group has a quotient which is this quotient and if you just take ideals Co Prime to M modulo principle just copyright that's the ordinary class group so all these Ray class groups they suggest to the ordinary class group they're just bigger versions and how much bigger are they well you have to see if you take a principal ideal what it is modulo M well that's just what you do you take your ring of integers oh modulo m you take the generator modulo unit because it has to be an ideal and of course it's all mod m star you know what it means if m is just an ideal it's the residue class ring you take its units this formal collection of real embeddings means that almost ammo star is just the thing itself m0 star but for the infinite primes dividing your modulus you take a side map so you take a sine P dividing and infinity so for real primes you also look at the quotient modulo the finite part and here we take the sign under the embeddon so it's a bit of a funny group yes certainly but I'm not proving anything I just claim things are true it is certainly true that you can generate any class group with the primes outside whatever sure yeah thank you all right so here are the new new zmod MC stars and the corresponding Fields so every n if you look at the theorem I'm about to wrap up foreign extension is just a subfield of array class field so every m does give rise to some field hm is the Ray Glass field modulo m for which the Galva group is exactly the ray class group so the equivalent so for if you write down your favorite modulus look at this this group it actually does arise as a Galva group namely of the ray class field modulo m so the question now becomes how do we find these Ray class group what are these are the analogs of the cycloatomic fields and the main problem is that they do not come with explicit generators as they did in the case of the rational number of fields the course which is starting in whatever 15 minutes by Yong vong and general rosu will do complex multiplication and they will solve this problem if K is imagery quadratic and in general it's an open Hilbert problem and in many ways this is not a problem since why would you need generators typically you only need the arithmetic of the number field and these groups that you just Define on the side of K give you all the control that you need I mean we still want to know but in many cases you don't care about generators you can still deal with it and once you know a Galva group you can use it with the chair with the roof theorem to say how things distribute over the Galva group so you're always in business as long as you're dealing with a building extensions so that is the key statement about classful Theory a building extensions of a number field are class Fields meaning that they're splitting Behavior the way their arithmetic is sort of set up it only depends on p modulo m analytically conclude in three or four minutes to prepare you for the exercises this afternoon classroom theory is something that you can actually use to solve problems and the problem that is the topic of a book by David Cox is that you look at the primes P P stands for Prime that are of the form x square plus and Y squared that seems like a very specific problem and this is a set of prime numbers called SN and the question is can you characterize these Primes and as you see if you take your favorite easiest value so n is going to be a positive integer and for this is a question that has been answered by Firma primes that are of the form x squared plus y Square are just the primes that are congruent to one module of four which means that you know these primes you also know how many there are if you believe dirichlet at least primes one mode four that's just half of the Prime so the density Delta n is one half easy theorem not in from us days maybe he had to use descent but nowadays modern themes it's easy S2 is an exercise this afternoon and also three and four that's easy S5 is the exercise that you find in my diagram since the reason that this has something to do with classical theory is that what you're asking here is that you're looking at the ring Z square root minus n and rather than looking at your Prime in P you're asking something about it's splitting behavior in this quadratic order then what you want is that your P decomposes as X Plus y squared minus n x minus so this is just saying that P decomposes as Pi times pi Bar for a prime element Pi in this quadratic order and for small values of M like two you may be lucky this is a ring inside the number field Q square root minus n and if it happens to be the Ring of integers you're in business you only need the classical algebraic number Theory text and you can actually determine how Prime splits so from -1 it's indeed just being one mode four will make it split split completely but you may know that for Z square root of minus five there is this if you take this for your ground field the class group interferes this is the popular example of a field or of a ring which is not a unique factorization domain I'm almost done don't get upset there's five into the coffee break but these five minutes will save them half an hour for the exercises so we're actually gaining a lot of time by e so that means that if you want to have primes with the form x squared plus five y Square then you wanted your Prime splits in z square root minus 5 which is the ring of integers however this is not a cloud number one field which means that you have a second condition not only does it need to split in this quadratic field Q squared minus five you also want primes to be actually principle and that's also a class but it's a class in the class group not a mod 5 kind of thing but a class like this so you see that in this case if you take k equal to Q squared minus 5 and M just equal to one no primes dividing it then already there's an interesting statement so for m is one you see that the class group of a number field corresponds to the Galva group of Summer being extension whatever h over K which is the Hilbert class field and I didn't get to all the formal properties of this art and map in which primes ramaphi but as you can imagine tomorrow we're going to see that the primes that ramify and allow for cable of course be the divisors of M and if M has no divisors this has to be totally unremified a billion and structure field exists for Q squared minus five and it's actually staring you in the face here it is this quadratic extension is the age that you need and in this case you actually prove it it is still not completely trivial it's one of the exercises that is of course that two groups of order two are isomorphic is not so shocking the fact is that the shocking thing is that the map is given by the art in map so the class of a prime in the class group of Q squared minus five will make will determine whether it splits or not in this specific quadratic extension well there's the miracle of classical Theory and I also see immediately that if you want to do this for arbitrary and already for four four is a square right if this n is not Square free you get an order in the maximum order and you should wonder what the class group of that order is in relation to the groups that you have in classroom Theory well since my time is up I cannot tell you right now but you'll find out this afternoon or you ask the assistants and then tomorrow we'll have someone telling us because Ricardo Pango is here on the second row is really important his duty I think is to convince volunteers to uh present problems during the what's it called exercise discussion session or something so of course you're completely free but this is a really a southern French style course so your orders are being taken from Ricardo you hired Italians to make sure you behave okay that was a break there's no break sorry we're gonna have young okay thank you for your attention [Applause] foreign