THE CHEBOTAREV DENSITY THEOREM

Transcription

1 THE CHEBOTAREV DENSITY THEOREM NICHOLAS GEORGE TRIANTAFILLOU Abstract. The Chebotarev density theorem states that for a finite Galois extension of number fields and a conjugacy class C Gal(), ] if δ denotes the density of rimes of K such that the Artin symbol satisfies = C then δ exists and is equal to #C/# Gal(). This result (and its roof) has layed an imortant role in the develoment of modern number theory, insiring Artin s roof of class field theory and several imortant results in modern arithmetic geometry. We give two roofs of the Chebotarev density theorem: Chebotarev s original roof Tsc26], which redates class field theory, and a more recent roof due to Lagarias and Odlyzko LO77] and refined by Serre Ser8] which gives an effective error term. We also give several alications of the Chebotarev density theorem and discuss its connection with Dirichlet s theorem on rimes in arithmetic rogressions. This work was comleted while suorted by an NSF Graduate Fellowshi. Contents. Introduction.. Outline 2 2. Statement of Main Results 3 3. The Connection to Dirichlet s Theorem on Primes in Arithmetic Progressions 5 4. Dirichlet-Hecke L-Functions and Artin L-Functions 6 5. The Chebotarev Density Theorem for Dirichlet Density 8 6. Effective Chebotarev for Natural Density 5 7. Alications of the Chebotarev Density Theorem 20 Aendix A. Notation 23 References 23. Introduction This aer is an exository aer on the Chebotarev density theorem and some alications written as a final roject for Math 229X - Introduction to Analytic Number Theory as taught by Noam Elkies at Harvard in the Fall of 205. We make an effort to kee this aer mostly self-contained, but in the interest of brevity, assume some standard definitions from analytic number theory. The notation used in the aer should be mostly self-exlanatory. We include a brief aendix on notation, just in case. Before we can discuss the Chebotarev density theorem, we need a definition.

2 2 NICHOLAS GEORGE TRIANTAFILLOU Definition. (The Artin Symbol). Let be a finite Galois extension of number fields. ] For any rime of O K and any rime P of L lying over, the Artin symbol is the unique element σ Gal() such that P (.) σ(α) = α Nm() (mod P) for all α L. For any, all of the rimes ] P lying over are isomorhic via elements of Gal(). Hence, the values of lying over are all conjugate in Gal(). We denote P ] ] by the conjugacy class of in Gal() for any P lying over. In the P ] case that Gal() is abelian, we abuse notation and use to refer to the single element of the conjugacy class. Informally, the Chebotarev density theorem says that for any finite Galois extension of number fields and ] any conjugacy class C Gal(), the number of rimes of O K such that = C is roortional to the size of C. This can be thought of as an equidistribution result on rimes. Although it is not obvious at first, we shall see throughout this aer (and articularly in Section 3) that the Chebotarev density theorem is a generalization of and is intimately related to Dirichlet s theorem on the infinitude of rimes in arithmetic rogressions. It would be difficult to overstate the imortance of the Chebotarev density theorem to modern number theory. Since its roof, it has been one of the most imortant tools for roving that various interesting subsets of rimes are infinite. In the ast 40 years, since the develoment of effective versions, the Chebotarev density has also been alied to the study of coefficients of modular forms, l-adic reresentations of infinite Galois grous, and many other subjects at the forefront of modern number theory and arithmetic geometry. Serre develoed several of these alications in Ser8]. These alications are beyond the scoe of this reort, so we content ourselves to very briefly describe the general flavor of these results in Section 7. The Chebotarev density theorem (or at least its original roof) had another extremely imortant consequence for modern number theory. While it might not be obvious from most modern treatments, which resent the density theorem as a consequence of class field theory, the roof of the Chebotarev density theorem redated and insired Artin s original roof of class field theory. For more details on the historical context of the Chebotarev density theorem in number theory (as well as an engaging account of the life and other work of Nikolai Chebotarev), see SL96]... Outline. The remainder of the aer begins with three sections of setu and general discussion, followed by two sections roving two different strengths of the Chebotarev density theorem by two different methods, and finishing with a section resenting some alications. More recisely, in Section 2, we give three different statements of the Chebotarev density theorem: Chebotarev s original statement without an error term and two effective versions due to Lagarias and Odlyzko LO77] with a refinement by Serre Ser8] that secify the error term of the Chebotarev density theorem for in terms of d L

3 THE CHEBOTAREV DENSITY THEOREM 3 (the absolute value of the the discriminant of L) and n L (the index of L over Q). After giving these recise statements of the Chebotarev density theorem, we demonstrate the relationshi between Chebotarev s density theorem and Dirichlet s theorem on rimes in arithmetic rogressions in Section 3. In Section 4 we introduce and briefly comare Artin L-functions and Dirichlet-Hecke L-functions, the central analytic objects needed for our roofs. Sections 5 and 6 contain most of the heavy analytic number theory. In Section 5, we resent a roof of the Chebotarev density theorem for Dirichlet density without error term. We roughly follow the treatment of Chebotarev s original roof as described in FJ08]. In articular, we do not assume knowledge of class field theory with the excetion of one statement the secial case of cyclotomic extensions. Section 6 is an exosition of Lagarias and Odlyzko s effective version of the Chebotarev density theorem, based on their roof in LO77]. We attemt to cature the heart of the argument, which is remarkably similar to the roof of Dirichlet s theorem in Dav00]. We fill in a few details omitted in LO77], while omitting other details covered in deth in the original aer. LO77] is an excellent resource for the reader interested in an extremely clear treatment of the details we omit. Finally, we close the aer by discussing several different roblems to which the Chebotarev density and its effective versions can be alied. We resent four different tyes of alications, including roving the infinitude of a subset of rimes, imroving uer bounds on the size of zero density sets of rimes, comuting Galois grous, and roving that different collections of data about field extensions are equivalent. Besides giving a bit more information on each of these alications in Section 7, we work out two simle alications in detail. 2. Statement of Main Results In order to state our main results recisely, we need to recall the following definitions. Definition 2. (Dirichlet and Natural Density for Number Fields). Let K be a number field, and let Q(K) be some set of rime ideals of O K. The quantity lim / (2.) s + (Nm ) s (Nm ) s Q(K) P (K) is called the Dirichlet density of Q(K) if the limit exists. The quantity (2.2) lim #{ Q(K) : Nm x}/#{ P (K) : Nm x} x is called the natural density of Q(K) if the limit exists. We can now state the main results of this aer. (The statements are quoted from FJ08] for Theorem 2.2, Ser8] for Theorem 2.3, and LO77] for Theorem 2.4.) Theorem 2.2. Let be a finite Galois extension of number fields and let C be a conjugacy class of Gal(). Let { ] } P C := P (K) : is unramified in L, = C.

4 4 NICHOLAS GEORGE TRIANTAFILLOU Then, the Dirichlet (res. natural) density of P C in { P (K) : is unramified in L} exists and is equal to #C # Gal(). Theorem 2.2 (for Dirichlet density) is essentially the original statement of the Chebotarev density theorem, as roved in Tsc26] in 926. Set (2.3) π C (x, ) = {# P C : Nm x}. After a short comutation, the natural density version of Theorem 2.2 can be rehrased as #C x (2.4) dt π C (x, ) # Gal() 2 log t #C x as x. # Gal() log x While this is certainly a owerful statement, early roofs of Theorem 2.2 and (2.4) either gave no information about the error terms or gave estimates including constants deending in an unredictable way on the fields L and K. In their 977 aer LO77], Lagarias and Odlyzko made rogress towards resolving this deficiency in the understanding of the subject by roving the following effective versions of the Chebotarev density theorem. We first give a statement conditional on GRH as refined by Serre in Ser8]. Theorem 2.3 (Theorem.. of LO77] refined as in Ser8]). Let be a Galois extension of number fields. Assume the generalized Riemann hyothesis for the Dedekind zeta function ζ L (s). Then, there is an effectively comutable ositive absolute constant c such that for every x > 2, (2.5) π C(x, ) #C # Gal() x 2 ( ) dt log t c #C # Gal() x/2 log(d L x n L ). Lagarias and Odlyzko also roved an unconditional version of their effective Chebotarev density theorem, with the remaining ineffectivity quantified in terms of the location of a ossible Siegel zero. The shae of their result is reminiscent of the quantitative versions of Dirichlet s theorem on rimes in arithmetic rogressions. The result is Theorem 2.4 (Theorem.3. of LO77]). If n L >, then ζ L (s) has at most one zero in the region defined by s = σ + it with (2.6) (4 log d L ) σ, t (4 log d L ). If such a zero exists, it must be real and simle. Call it β 0. Then, there exist effectively comutable absolute constants c and c 2 such that if (2.7) x ex(0n L (log d L ) 2 ), then (2.8) π #C x dt C(x, ) # Gal() 2 log t #C # Gal() x β 0 2 dt ( ) log t + c x ex c 2 n /2 L (log x) /2,

5 THE CHEBOTAREV DENSITY THEOREM 5 where the β 0 term is resent only when β 0 exists. If C is relaced by a union of conjugacy classes, the same estimate holds after multilying the last error term by the number of conjugacy classes in C. Theorems 2.3 and ] 2.4 lead immediately to uer bounds on the lowest norm of a rime with = C, although better results along these lines can be achieved via more secialized methods, as in LMO79]. While there are several other statements of the Chebotarev density theorem, including a version for function fields (see for examle FJ08]), such further statements are beyond the scoe of this aer. 3. The Connection to Dirichlet s Theorem on Primes in Arithmetic Progressions While on the surface the Chebotarev density theorem is a statement about how rime ideals behave under field extensions, the Chebotarev density theorem is also a generalization of Dirichlet s Theorem on rimes in arithmetic rogressions. Indeed, Dirichlet s theorem (modulo n) is the secial case of the Chebotarev density theorem when K = Q and L = Q(ζ n ), where ζ n is a rimitive nth root of unity. Corollary 3. (Dirichlet s Theorem on Primes in Arithmetic Progressions). Let a, n be ositive integers with (a, n) =. Then, the natural (res. Dirichlet) density of rimes such that a (mod n) in the set of all rimes of Z is φ(n). Proof. Let ζ n Q be a rimitive nth root of unity, K = Q, and L = Q(ζ n ). By the theory of cyclotomic extensions, is Galois with Gal() = (Z/nZ). This isomorhism is given exlicitly by identifying an element a (Z/nZ) with the unique automorhism of that mas ζn k to ζn ak for all k. Now, for a rime number Z, Nm(Z) =. If P is a rime of O L lying over such that σ Gal() satisfies σ(α) = α Nm(Z) mod P, we must have that σ(ζn) k ] = ζn k mod P for all k. So long as n, the ideal Z does not ramify in L, so = Z ] Gal(), where is the class of modulo n. Thus, = a if and only if a Z mod n. In our setting, Theorem 2.2 states ] that the natural (res. Dirichlet) density of rimes ideals Z of Q such that = a is =. In other words, the natural Z # Gal() φ(n) (res. Dirichlet) density of rimes such that a mod n is. φ(n) We have seen that Dirichlet s theorem is a secial case of Chebotarev s density theorem. In some sense, Dirichlet s theorem is much more than a secial case. For instance, the main analytic ste in the roof of the Chebotarev density theorem is used to rove the theorem in the case where is a cyclotomic extension (or an abelian extension if class field theory is assumed). The strategy for roving Chebotarev in this secial case is very similar to the roof of Dirichlet s theorem, with Dirichlet L-functions relaced by a suitable generalization defined in terms of rime ideals of O K and the extension

6 6 NICHOLAS GEORGE TRIANTAFILLOU L. The full ower of Chebotarev, first for abelian extensions and then for general extensions, follows by some clever alications of Galois theory. Even when controlling the error term, the strategy of roof is exactly the same as with Dirichlet s theorem, but with more comlicated book-keeing. Modern roofs taking into account the error term tend to closely mirror the roof of Dirichlet s theorem with error term. 4. Dirichlet-Hecke L-Functions and Artin L-Functions Having seen that the Chebotarev density theorem is a generalization of Dirichlet s theorem, we would like to mimic the roof. Before we have any hoe of carrying out the roof, we need to answer the question: What is the right generalization of Dirichlet L-functions? For our uroses, there are two answers: Dirichlet-Hecke L-functions and Artin L- functions. In the remainder of this section, we define both tyes of L-functions and discuss how they are related. First, we define Dirichlet-Hecke L-functions. These L-functions generalize Dirichlet L-functions to base fields other than Q. We first need to set u some notation as in section VI of Lan86]. Definition 4.. Let K be a number field with ring of integers O K. Let c = m() be an ideal of O K with its factorization into rimes. Denote the localization of O K at by O with maximal ideal m. Set (4.) I(c) := {a fractional ideals of K : a and c are relatively rime.}, with the natural grou structure and let (4.2) P (c) := {rincial fractional ideals xo K of K satisfying () and (2).} () If m() > 0, i.e. c, then x O and moreover x mod m m(). (2) For each real embedding σ : K R, σ(x) > 0, i.e. x is a totally ositive element of K. The quotient G(c) = I(c)/P (c) is called the c-ideal class grou of c. Remark 4.2. To be recise, we are really defining the ideal class grou of the c together with all of the real laces. The condition on the real embeddings can be relaxed to get an ideal class grou which only takes into account some of the real laces. As with the usual ideal class grou, the c-ideal class grou is also finite. Theorem of chater VI of Lan86] rovides an exlicit formula for the cardinality. The finite abelian grou G(c) will lay the role in the construction of Dirichlet-Hecke L-functions that the grou (Z/nZ) lays in the construction of Dirichlet L-functions. Indeed, G(c) is a generalization of (Z/nZ) to ideals of number fields. If K = Z and c = nz, then I(nZ) = {az : a Z, (a, n) = } and P (nz) = {az : a Z, a (mod n)}, so G(nZ) = (Z/nZ). Let Ĝ(c) denote the grou of characters of G(c). We are now ready to define the Dirichlet-Hecke L-function associated to χ Ĝ(c).

7 THE CHEBOTAREV DENSITY THEOREM 7 Definition 4.3. Let c be an ideal of O K and let χ Ĝ(c). The Dirichlet-Hecke L- function associated to c and χ is the analytic function L c (s, χ) := χ(a) (Nm a) = ( χ() ) (4.3), s (N) s (a,c)= (,c)= defined on the half-lane Re(s) >, where the sum is over all ideals a O K relatively rime to c and the roduct is over all rime ideals O K relatively rime to c. Note that the Euler roduct and L-series are equal because χ(a) χ(b) = χ(a b). Next, we define Artin L-functions, our second generalization of Dirichlet L-functions. Artin L-functions are defined as an Euler roduct involving reresentations of Gal() for a finite Galois extension of number fields. Definition 4.4. Let be a Galois extension of number fields with Galois grou Gal(). Let ρ : Gal() V be a finite dimensional comlex reresentation. For each rime of O K, choose a rime P of O L lying over. Then, the Artin L-function associated to and ρ is L (s, ρ) := ( ( ])) (4.4) det (Nm ) s ρ, P O K on Re(s) >, where we restrict to the subsace of V of elements fixed by the inertia grou at P before taking the determinant if ramifies in L. Note that the local factors are well-defined irresective of the choice of P over since making a different choice of P corresonds to conjugating the matrix the determinant is conjugation-invariant. ( (Nm ) s ρ ( P ])) by some ρ(σ) and In the secial case where L = Q(ζ n ) and K = Q, we can identify elements of Gal() with characters of (Z/nZ). In this case, the local factors at the unramified rimes are ( χ() s ), and (at least for rimes not dividing n) we recover the Dirichlet L-functions modulo n. Remark 4.5. Throughout the remainder of this aer, we will tyically ignore the ramified rimes when discussing Artin L-functions, although we will oint out/bound their contribution when necessary. While this may seem cavalier, very little will be lost for our uroses since we are interested in density statements. Only finitely many rimes ramify in a finite extension of number fields and the ossible rimes are wellcontrolled by invariants of the field, so ramification won t interfere too badly with our estimates. Although Artin L-functions cannot tyically be exressed as L-series when Gal() is non-abelian, the logarithm and logarithmic derivatives of Artin L-functions still have nice exressions. (See Section 2..4 of Sny02] for full details.) Since L (s, ρ) is defined as an Euler roduct, log L (s, ρ) is a sum over local factors. At a rime ( ( ])) which does not ramify in L, the factor is log det (Nm ) s ρ. Diagonalizing ρ turns the determinant into a roduct. Taking Taylor exansions P ]) of ( P

8 8 NICHOLAS GEORGE TRIANTAFILLOU each term, the factor at is m= ( Tr ρ m P ] m ) (Nm ) sm. The ramified rimes require a bit more attention since the local factors are more comlicated. The result is as follows: First, if φ = Tr(ρ) : Gal() C is the character associated to the reresentation ρ : Gal() V, P is any ideal of O L lying over the ideal of O K, and I P is the inertia grou at P, write (4.5) This is well-defined since function. In this notation, (4.6) (4.7) φ K ( m ) := #I P σ(p) log L (s, φ) = α I P φ ] m = σ P O K m= L L (s, φ) = O K m= ( P ] m α). ] m σ, I σ(p) = σi P σ and φ is a class m φ K( m )(Nm ) ms, φ K ( m ) log(nm )(Nm ) ms. Warning: the functions φ K are in general non-linear. In articular, we cannot take the exonent outside of the φ K. Dirichlet-Hecke L-functions and Artin L-functions each have their own advantages and disadvantages, articularly in the context of the Chebotarev density theorem. Artin L-functions clearly involve the Artin symbol the function we are studying. The downside is that Artin L-functions use non-linear characters of the Galois grou. As a result, we no longer have an L-series reresentation and therefore are derived of several standard tools. For instance, the roof of analyticity for Dirichlet L-functions does not hold for general Artin L-functions. When working with Dirichlet-Hecke L-functions, the toolbox is full, but it is less clear how characters of a c-ideal class grou relate to the Artin character. Fortunately, class field theory imlies that when is an abelian extension, Artin L-functions are the same as Dirichlet-Hecke L-functions for aroriately chosen c (u to modification at the ramified laces). The general case of the Chebotarev density theorem follows from the case of abelian (in fact, cyclic) Galois grou, using only a bit of field theory and reresentation theory of finite grous. Hence, we may assume that the extensions we are dealing with are abelian, in which case, we have the best of both worlds. At this oint, we recall that although this view may suggest that the Chebotarev density theorem is a consequence of class field theory, Chebotarev did not have access to the full statements of class field theory when he wrote his aer Tsc26] in 926. Indeed, the Chebotarev density theorem led to the first full roofs of class field theory. 5. The Chebotarev Density Theorem for Dirichlet Density In this section, we rove Theorem 2.2, the original statement of the Chebotarev density theorem for Dirichlet density, following Chebotarev s original strategy Tsc26].

9 THE CHEBOTAREV DENSITY THEOREM 9 Our treatment is insired by FJ08], although our notation is slightly different. The roof starts with the case that is cyclotomic. This case goes as follows: () Comute the volume of a fundamental domain of a certain lattice to bound the number of reresentatives of a class in G(c) with small norm. Use this to meromorhically continue L c (s, χ) to a function on Re(s) > n K which is analytic, excet for a ossible ole at which exists if and only if χ is trivial. (2) Prove a result from class field theory for cyclotomic extensions to see that the Dirichlet-Hecke L-functions are Artin L-functions. (3) Use class field theory for cyclotomic extensions and the behavior at s = of the zeta function L col (s, χ 0 ) to conclude that L c (s, χ) 0 for χ χ 0. (4) Take an aroriate character-weighted linear combination of the L c (s, χ). Note that stes (), (3), and (4) are analogous to stes in the roof of Dirichlet s theorem without error term. The case that is abelian follows from the cyclotomic case by Chebotarev s field crossing argument. The general case can be reduced to the cyclic (and therefore abelian) case using a bit of reresentation theory. Proof of Theorem 2.2 for Cyclotomic Extensions. We first recall that the abscissa of convergence of an L-series is controlled by its coefficients. Secifically, Proosition 5. (Lan86], Ch. VIII Theorem 4). Let {a n } be a sequence of comlex numbers, with artial sums A n. Let 0 σ 0 < and suose that for some ρ C, A n = nρ + O(n σ 0 ). Then, f(s) = a n /n s has an analytic continuation to Re(s) > σ 0, excet for a simle ole with residue ρ at s = when ρ is non-zero. Proof. If ρ is non-zero, let b n = a n ρ, let B n be the artial sums of b n, and let g(s) = b n /n s. Then, f(s) = g(s)+ρζ(s). Write s = σ +it. If σ > σ 0, by summationby-arts, N n=m+ b N+/2 n n = s M+/2 N σ σ 0 + s x db(x) = B s N N+/2 M+/2 so the series clearly converges at s. N+/2 (N + /2) + s s M+/2 B(x) dx xs+ x σ+ σ 0 dx N σ σ 0 + s (σ σ 0 )(M + /2) σ σ 0, To take advantage of this roosition, we rewrite the L c (s, χ) as a sum over classes in G(c). We use the following roosition, which aears as Theorem 3 of Chater VI of Lan86]. Proosition 5.2. Let c be an ideal of O K and A be a class in G(c). There is a constant ρ c deending only on c such that the number of ideals a of O K such that a A and Nm a n is ρ c n + O(N n K ). We omit the roof, which involves comuting the volume of the fundamental domain of a lattice.

10 0 NICHOLAS GEORGE TRIANTAFILLOU Since ρ c is indeendent of the class A G(C), breaking u the sum of the coefficients χ(a) of L c by class shows that as n, {#G(c) ρ c n + O(n n K ), χ = χ0 (5.) χ(a) = O(n n K ), χ χ0. (a,c)=,nm a n Alying Proosition 5. roves Lemma 5.3. L c (s, χ) extends to a meromorhic function on Re(s) > n K analytic excet for a simle ole of residue #G(c)ρ c at when χ = χ 0. which is Now that we have roved an analytic continuation, we wish to move toward the world of Artin L-functions. We could do this for all abelian extensions by citing an aroriate theorem of class field theory. However, in the cyclotomic case, the roof has an interesting analytic ste, so we resent it below. Suose for now that is an abelian extension, and let c be a rime such that c for all rimes of K that ramify in L. Then, the Artin symbol extends by multilication to a homomorhism I(c) Gal(), which we denote the case where is cyclotomic, we have: ]. Restricting further to Lemma 5.4 (Lemma of FJ08]). Let ζ m be a rimitive mth root of, let be a Galois extension of number fields] such that L K(ζ m ) and let c be an O K -ideal such that m c. Then, P (c) ker. c The roof is not hard but is comletely algebraic, so we omit it. See FJ08] for details. ] ] Lemma 5.4 imlies that descends to a ma : G(c) Gal(). ] c c Let G := image Gal(). Given a character χ Ĝ, the comosition ] c χ is a character of G(c). This allows us to write L col (s, χ 0 ) in terms of the c ] ) ] L c (s, χ. Working at the level of local factors, for c, the order of is c equal to the inertial degree f. Since does not ramify in L, there are # Gal() f lying over. Hence, (5.2) P lying over ( (Nm P) = s (Nm ) sf ) # Gal() f c = χ Ĝ χ ( rimes ]) (Nm ) sf where the second equality is a standard statement about characters. Taking the roduct of both sides over all rime to c, gives the formula (5.3) L col (s, χ 0 ) = ] L c (s, χ χ Ĝ c ) # Gal() #G. # Gal() #G,

11 Using this formula, we see THE CHEBOTAREV DENSITY THEOREM Theorem 5.5. Let ζ m Q be a rimitive mth root of, let be a Galois extension of number fields satisfying K L K(ζ m ) and let c K be an ideal divisible by m. ] Then, the ma : G(c) Gal() is surjective. c ] ) Moreover, if χ is a nontrivial character of Gal(), L c (, χ 0. c Proof. For any character χ Ĝ, Lemma 5.3, imlies that ] ) { a χ, χ χ 0 (5.4) ord s= L c (, χ =, χ = χ 0, for some constants a χ Z 0. Alying (5.4) to (5.3) yields (5.5) = # Gal() #G c + a χ. χ Ĝ Then, # Gal() divides, so G = Gal(). Since a #G χ 0 for χ χ 0, we must have a χ = 0 for χ χ 0. The statement Gal() = G means that the Galois characters all induce characters of G(c), so the Artin L-functions for are all Dirichlet-Hecke L-functions for c. Having roved the secial case of class field theory that we needed, we are ready to begin our final attack on the Chebotarev density theorem for cyclotomic extensions. Recall that to aly Dirichlet L-functions to comute the density of rimes in Q congruent to a mod n, one first shows that the contribution of higher owers in the double summation exression for log L(s, χ) is O() as s from above and then takes a linear combination of the log L(s, χ) weighted by aroriate characters. We do the same here. Lemma 5.6. Let ζ m Q be a rimitive mth root of unity, let be a Galois extension of number fields such that K L K(ζ m ), and let χ be a character of Gal(). Then, as s +, ( ]) log L (s, χ) = χ (5.6) + O(). (Nm ) s O K Proof. From the Euler roduct and the Taylor series for log, we know that ( ] ) k log L (s, χ) = χ (5.7) + O(), k(nm ) sk O K k=

12 2 NICHOLAS GEORGE TRIANTAFILLOU where the O() term accounts for the finitely many ramified rimes. There are at most n K rimes of O K lying above any rime Z Z and each has norm at least. Hence, the total contribution from the k > is bounded in magnitude by (5.8) which roves the lemma. n K k=2 O(), kσ Since the ramified laces affect only the residue of the ole of L (s, χ) at and not its order, given any σ Gal(), Lemma 5.3 imlies that as s +, (5.9) (5.0) χ Gal() log L (s, χ 0 ) = log(s ) + O(), and # Gal() χ(σ ) log L (s, χ) = # Gal() log(s ). On the other hand, by the orthogonality of characters and Lemma 5.6, (5.) log L (s, χ 0 ) = (Nm ) + O(), s (5.2) χ Gal() # Gal() χ(σ ) log L (s, χ) = O K O K : ]=σ Putting these together, we see that lim s O + = K (Nm ) s the roof of the Chebotarev density theorem for cyclotomic extensions. O K : ]=σ (Nm ) s and (Nm ) s + O(). L : K], comleting Proof of Theorem 2.2 for Abelian Extensions. To extend from the cyclotomic case to the general abelian case, we have two otions. One otion is to cite the aroriate theorems of class field theory to show that for any abelian, we can still choose an ideal c O K such that the ma G(c) Gal() is surjective. The remainder of the roof from the cyclotomic case then goes through for the abelian case nearly unchanged. Instead, we resent a field crossing argument similar to that of Chebotarev s original roof. Related ideas are often used as a reliminary ste in roofs of class field theory, including Artin s original roof. We start by outlining the roof, generally following the aroach of FJ08]. Given an abelian Galois extension of number fields and σ Gal(), the first ste is to construct an auxiliary cyclotomic extension of number fields F/E of degree k order(σ) for k Z such that L F, K E, and F/K is Galois. The contribution of rimes of E which are not slit in E/K can be ignored, since there are finitely many ramified rimes and the rest have large norm. This allows us to comute the densities for many subsets of elements of Gal(F/K) which restrict to σ on L. The sum of these densities gives a lower bound for the density of the rimes maing to the conjugacy class C Gal() of σ. Taking an aroriate sequence of fields gives the correct lower bound. Observing that all of the densities sum to comletes the roof. We now give the argument in full detail.

13 THE CHEBOTAREV DENSITY THEOREM 3 ] = σ. To start, we bring We wish to comute the density of rimes such that in an auxiliary cyclotomic field extension. Choose a cyclotomic extension M/K such that M/K has an element τ with ord σ ord τ and M L = K. Then, the comositum F = LM is Galois over K with Gal(F/K) = Gal() Gal(M/K). Let ρ = (σ, τ) Gal(F/K). Note that ord ρ = ord τ by our choice of τ. Let E = F ρ be the fixed field of ρ. Then, E M = M τ. Thus, M : E M] = ord τ = ord ρ = F : E] and so F = EM. Thus, F is a cyclotomic extension of E. By the Chebotarev density theorem for cyclotomic extensions, the set Q (ρ) of rimes q O E such that ] O E such that F/E q F/E q ] = ρ has density F :E]. Denote by Q 2(ρ) the set of rimes q = ρ and q has degree above = q K. Since the sum Nm q over all rimes of E which do not have degree above is bounded by q ramified + E : Nm q K], the density of Q 2 2 and Q are equal. Also, if] q Q 2 (ρ) ] and = q K, then Nm q = Nm. It follows immediately that ρ = =. Moreover, any rime lying under some element of Q 2 (ρ) must F/E q F/K have exactly E : K] rimes in Q 2 (ρ) lying over it by the definition of Q 2 (ρ). Since elements q Q 2 (ρ) satisfy ] Nm L q = Nm K (K q), this imlies that the set P ρ of rimes O K such that F/K = ρ has density at least =. E:K] F :E] F :K] Now, if P ρ, ρ(α) = α Nm mod P for all α F for any P O F lying over, so ρ(α) = α Nm ] O K : mod P L for all α L, which imlies that the density of P σ = { = σ} has density at least F :K] = L:K] M:K]. We have already shown that P σ has ositive density. To go further, note that we were free to choose any τ Gal(M/K) so long as ord(σ) ord(τ). For different choices of τ, the sets P ((σ, τ)) will be disjoint, so the sums of their densities will also be a lower bound on the density of P σ. Now, if ord σ = α α k k and Gal(M/K) is cyclic of order β β l, the number of elements of of Gal(M/K) of order a multile of order(σ) is l (5.3) ( β α ) ( β k k α k k ) β k+ k+ β l l = M : K] k ( α j β j j ). j= Assuming we can construct such an M, the density of P σ is at least (5.4) L : K] k ( α j β j j ). j= Of course, by Dirichlet s theorem on rimes in arithmetic rogressions, i.e. the secial case of the Chebotarev density theorem for the cyclotomic extension Q(ζ m )/Q, there are infinitely many rimes such that (mod m) for any m = j β j j. For any sufficiently large such, K(ζ ) L = K and Gal(K(ζ )/K) = (Z/Z) = Z/( )Z. Thus, we can choose a sequence of such fields M = K(ζ ) so that the β j get arbitrarily large. Thus, the Dirichlet density of P σ is at least. This argument alies to all L:K]

14 4 NICHOLAS GEORGE TRIANTAFILLOU σ Gal(), the P σ are disjoint, and the total Dirichlet density is, so in fact, the Dirichlet density of P σ is exactly. L:K] Proof of Theorem 2.2 for Finite Galois Extensions. For this section, we deart from FJ08], which gives a more combinatorial argument reducing the roof to the case of cyclic extensions in favor of the reresentation theory based argument in Section 4 of LO77]. The general case follows from the cyclic case by the reresentation theory of finite grous. Given a finite Galois extension Gal() and a conjugacy class C Gal(), ick some σ C and let E = L σ be the fixed field of σ. Then, Gal(L/E) = σ is the cyclic subgrou of Gal() generated by σ. Orthogonality of characters imlies that for τ Gal(), (5.5) φ, char. of Gal() φ(σ)φ(τ) = χ Gal(L/E) χ(σ) Tr Ind Gal() σ χ(τ) = { # Gal(), #C τ C, 0, τ / C. By (4.6), absorbing the finitely many ramified rimes and the contribution from m 2 into an O() term as before, we have that for any σ C, O K : ]=C (5.6) (Nm ) s #C + O() = # Gal() #C = # Gal() φ, char. of Gal() χ Gal(L/E) φ(σ) log L (s, φ) χ(σ) log L (s, Tr Ind Gal() Gal(L/E) χ). Now, it is not hard to check by Mackey s Theorem in the unramified case and an exlicit comutation in the case of ramification that L (s, Ind Gal() Gal(L/E) χ) = L L/E(s, χ), (see, for examle Section of Sny02]). Continuing from (5.6), (5.7) O K : ]=C (Nm ) s #C + O() = # Gal() = #C # Gal(L/E) # Gal() χ Gal(L/E) q O E : L/E q ]=σ as s +. Since the class {σ} has density in P # Gal(L/E) E and also, (5.8) (Nm ) s = (Nm q) s + O(), O K q O E as s +, we see that the Dirichlet density of rimes O K χ(σ) log L L/E (s, χ) (Nm q) s + O(), with ] #C # Gal(L/E) is = #C, which comletes the roof of the Chebotarev # Gal() # Gal(L/E) # Gal() density theorem for Dirichlet density. = C

15 THE CHEBOTAREV DENSITY THEOREM 5 6. Effective Chebotarev for Natural Density For many alications of the Chebotarev density theorem, Theorem 2.2 is sufficient. In articular, if the end-user s goal is simly to rove the infinitude of some set of rimes, there is no need to cite a full-strength Chebotarev. However, if one requires a rime (or many rimes) of relatively low norm in some ideal class, error bounds given in terms of numerical invariants of the fields are very useful. This is exactly what Theorems 2.3 and 2.4, roved by Lagarias and Odlyzko in LO77] and refined by Serre in Ser8], rovide. While these theorems did not give the world record bound on the lowest norm rime in an ideal class, (the record at the time still belonging to Lagarias and Odlyzko with the addition of Montgomery LMO79]), they were still the first examle of a roof of the Cebotarev density theorem with an exlicit, effectively comutable error term. We follow Lagarias and Odlyzko s extremely clear resentation of the roof from their original aer LO77], omitting all but a select few articularly instructive details. The outline of the roof could almost be read out of Davenort s Multilicative Number Theory Dav00], although there are a few additional wrinkles. (In articular, there will be some inuts from class field theory and reresentation theory. The error terms will also be more comlicated). The general argument goes as follows: () Comare ψ C (x, ) := Nm m x, ] m log(nm ) to a truncated inverse =C unramified Mellin tranform of a character-weighted linear combination of the logarithms of Artin L-functions. (2) Use an argument as in the roof of Theorem 2.2 for finite Galois extensions at the end of Section 5 to relace the Artin L-functions associated to with Artin L-functions associated to a cyclic extension L/E (which are then Dirichlet-Hecke L-functions by class field theory). (3) Use the functional equation and Hadamard roduct formula of the L-functions in question and the inequality cos θ + cos 2θ 0 to rove a zero-free (or nearly zero-free) region for ζ L (s) := χ L L/E(s, χ). Alternately, assume the generalized Riemann hyothesis for ζ L. (4) Use the functional equation and Hadamard roduct to bound the number of non-trivial zeros of the L-functions with imaginary art between t and t +. (5) Shift the line of integration from the truncated inverse Mellin transform to the left to get an exlicit formula for ψ C (x, ) with main term x #C/# Gal() and with the error term deending on a sum over zeros of L-functions. (6) Put everything together and choose an aroriate value of T (the value at which the Mellin transform was truncated) to balance error terms. The result is an estimate for ψ C (x, ) with effective error terms. (7) Use summation by arts to rove the corresonding result for π C (x, ). We now give a more detailed outline, filling in some details that Lagarias and Odlyzko leave unroven and including most of the key equations. We move quickly through the first coule of stes.

16 6 NICHOLAS GEORGE TRIANTAFILLOU Orthogonality of characters and (4.7) imly (3.5) and (3.6) of LO77], which state that for any τ C, (6.) #C F C (s) := # Gal() φ char. of Gal() where θ( m ) is an indicator function for takes values in 0, ] if ramifies in L. Let σ 0 = + / log x and let (6.2) I C (x, T ) := 2πi φ(τ) L L (s, φ) = O K m= θ( m ) log(nm )(Nm ) ms, ] m = C if does not ramify in L and σ0 +it σ 0 it F C (s) xs s ds be the truncated inverse Mellin transform of F C (s). ψ C (x) is the sum u to norm x of the coefficients of F C (s), with the ramified rimes omitted. Serre shows in Sections.3. and 2.4. of Ser8] that (6.3) O K,m:(Nm ) m x ramified in log Nm 2 log x O K ramified in log Nm 4 # Gal() log x log d L, which gives an uer bound on the error term resulting from the unramified rimes. Alying the same strategy for bounding the error term from the Mellin transform as in Dav00] s roof of Dirichlet s theorem yields the imroved version of (3.0) and (3.8) of LO77], (6.4) ψ C (x) I C (x, T ) log x log d L # Gal() + n K log x + n K x T (log x)2. Now, the same argument as in the reduction of Theorem 2.2 to the case of cyclic extensions shows that we may relace the sum φ char. of Gal() φ(τ) L L (s, φ) in the definition of F C (s) with χ Gal(L/E) χ(τ) L L/E L L/E (s, χ), where E is the fixed field of τ. We now recall some facts about the L L/E (s, χ). The L-functions L L/E (s, χ) have a comleted L-function ξ L/E (s, χ). ξ L/E has a functional equation ξ L/E ( s, χ) = W (χ)ξ L/E (s, χ). Also, ξ L/E, is entire of order. If ξ L/E is non-vanishing in s, then ξ L/E has a Hadamard roduct. Indeed, this is the case: Proosition 6.. L L/E (s, χ) has no zeros in σ. Hence the zeros of ξ L/E (s, χ) lie in the critical stri {0 < Re(s) < }. Proof. To show the result for L L/E (s, χ), it suffices to rove the roosition for ζ L (s), where ζ L (s) := L L/E (s, χ) = (6.5) (Nm P) = (Nm a) s. s P O L 0 A O L χ Gal(L/E)

17 THE CHEBOTAREV DENSITY THEOREM 7 From the last exression, ζ L (s) is clearly ositive for all s R with s > and has a simle ole with residue at s =. Then, since cos θ + cos 2θ 0, for any σ >, t R, taking the Taylor exansion of log( Nm P) for each P O L, (6.6) Re 3 log ζ L (σ) + 4 log ζ L (σ + it) + log ζ L (σ + 2it)] 0, and so ζ L (σ) 3 ζ L (σ+it) 4 ζ L (σ+2it). Now, for σ =, t = 0, ζ L (σ+it) has a ole and so is clearly non-zero. For t 0, if ζ L ( + it) = 0, then lim σ + ζ L (σ) 3 ζ L (σ + it) 4 = 0. Also, ζ L (σ + 2it) does not have a ole, so the whole roduct tends to zero. But this contradicts that the roduct has magnitude at least, so ζ L (σ + it) cannot have a zero with σ. The statement about ξ L/E (sχ) follows immediately from the functional equation. The logarithmic derivative of the Hadamard roduct for ξ L/E yields (6.7) L L/E (s, χ) = B(χ) + L L/E ρ:l L/E (ρ,χ)=0 ( s ρ + ) ( ρ 2 log A(χ) δ χ 0 (χ) s + ) γ χ (s), s γ χ where A(χ) is the roduct of d E with the norm of the conductor of χ, B(χ) is some constant deending on χ, and γ χ is a certain roduct of owers of π and Gamma functions coming from the infinite laces. We can now imrove the zero-free region from Proosition 6. by observing that the functional equation of ξ L/E (s, χ) imlies (6.8) ( s ρ + ) s ρ ρ ( = log A(χ) + 2δ χ0 (χ) s + ) + 2 γ χ (s) L L/E (s, χ) L L/E (s, χ). s γ χ L L/E L L/E Summing over all χ Gal(L/E) and alying a refined version of the argument in Proosition 6. with ζ L ζl in lace of log ζ L, gives the following zero-free and almost zero-free regions, which we state without roof. Lemma 6.2 (Lemma 8.. of LO77]). There is an effectively comutable ositive absolute constant c such that ζ L (s) has no zeros ρ in the region (6.9) Im(ρ) c, Re(ρ) + 4 log d L log d L + n L log( Im(ρ) + 2). Lemma 6.3 (Lemma 8.2. of LO77]). If n L >, ζ L (s) has at most one zero ρ in the region (6.0) Im(ρ), Re(ρ). 4 log d L 4 log d L If such a zero exists, it is real, simle, and comes from L L/E (s, χ) for a real character χ. Another imortant consequence of (6.8) and the functional equation of ξ L/E (s, χ) is Lemma 6.4, which bounds the number of zeros in a thin stri.

18 8 NICHOLAS GEORGE TRIANTAFILLOU Lemma 6.4 (Lemma 5.4. of LO77]). Let n χ (t) be the number of zeros of L L/E (s, χ) with imaginary art in the interval t, t + ] and real art in (0, ). For all t, n χ (t) log A(χ) + n E log( t + 2). Proof. At s = 2 + it, γ χ γ χ (s) n E log( t + 2), L L/E L L/E (s, χ) ζ E ζe (2) n E, and and s are both O(). Taking the real art of both sides of (6.8) yields s ( ) Re 2 + it ρ + (6.) log A(χ) + n E log( t + 2), 2 + it ρ ρ where we omit the absolute value on the left-hand ( side since ) every term is ositive. Now, if Im(ρ) t < and 0 < Re(ρ) <, Re 2+it ρ 2+it ρ, so the left hand 5 side of (6.) is n χ (t). This has the imortant consequence that the mysterious constant B(χ) satisfies B(χ) + (6.2) ρ log A(χ) + n E. ρ: ρ </2 See Lemma 5.5. of LO77] for a slightly more general statement and roof. The next ste is to use these bounds to show that if U > is a ositive half-integer and Π(U, T ) is the ath given by concatenating the straight lines from σ 0 +it to U +it to U it to σ 0 it, then (6.3) 2πi Π(U,T ) x s s L L/E (s, χ) x log x L L/E T (log A(χ) + n E log T ) + T x U U (log A(χ) + n E log(t + U)). The first term comes from the horizontal integrals and the second term comes from the vertical integrals. The only difficulty is bounding the contribution of the horizontal integrals near the critical stri. For this, it is helful to comare the values of L L/E L L/E (σ + it, χ) and L L/E L L/E (3+it, χ) to estimate L L/E L L/E (s, χ) by ρ: Im(s ρ)< within a log(a(χ))+ s ρ n E log( t + 2) error term. Now, the Cauchy integral formula and (6.3) imly that (6.4) 2πi σ0 +it σ 0 it x s s L L/E L L/E (s, χ) = ν oles of xs s L L/E L L/E (s,χ) ν U,σ 0 ]+i T,T ], ( ) x s L L/E res ν (s, χ) s L L/E with error term E as in (6.3). There are four sources for oles of xs s L L/E L L/E (s, χ): + E, () The non-trivial zeros of L L/E (s, χ). Each contributes a residue xρ (counting ρ zeros with multilicity). (2) The trivial zeros of L L/E (s, χ) at negative integers > U. The residue at k is a k (mod 2) (χ) x k, where a k k (mod 2)(χ) deends only on the arity of k and γ E (χ).

19 (6.5) THE CHEBOTAREV DENSITY THEOREM 9 (3) The ossible ole of L L/E L L/E (s, χ) at which contributes the residue δ χ0 (χ)x. (4) The ole of xs s and ossible ole of L L/E L L/E (s, χ) at 0. The residue here is some R(χ), which can be shown to satisfy R(χ) ρ log A(χ) + n E log(x), ρ: ρ < 2 by alying (6.2) and easy estimates on the size of L L/E L L/E (s, χ) and γ E γe (s, χ). (Note that our notation dearts a bit from LO77] here. In articular, LO77] breaks this term down further before deriving the bound.) Taking the limit as U, summing these terms over χ Gal(L/E), (weighted by χ(τ) of course), and alying the identity n L = L : E] n E and the conductordiscriminant formula χ log A(χ) = log d L, we arrive at Theorem 6.5. Theorem 6.5 (Theorem 7. of LO77]). Suose x 2 and T 2. Define #C S(x, T ) = χ(τ) x ρ # Gal() ρ (6.6), ρ χ Gal(L/E) ρ: Im(ρ) <T ρ: ρ < 2 where the sums are over zeros of ζ L (s). Then, ψ #C C(x, ) # Gal() x S(x, T ) + log x log d L # Gal() + n L x(log x) 2 # Gal() T ( #C x log x + T + log d L + n L log x + n ) Lx log x log T (6.7) # Gal() T T We are now ready to rove Theorem 2.3. Proof of Theorem Under GRH for ζ L (s), x ρ ρ ρ ρ: Im(ρ) <T = x ρ ρ ρ: ρ < ρ: Im(ρ) <T x/2 ρ ρ: Im(ρ) <T 2 T x /2 χ Gal(L/E) n χ(t) (6.8) x /2 log T (log d L + n L log T ). t t= Taking T x /2 to balance the error terms in (6.7) yields ψ #C C(x, ) # Gal() x #C (6.9) # Gal() x/2 log(d L x n L )(log x).

20 20 NICHOLAS GEORGE TRIANTAFILLOU Removing the terms for term of at most n K x /2 since (6.20),m 2:Nm K/Q () m x ] m = C for m 2 from ψc (x, ) introduces an error log Nm K/Q n K,m: m x log n K x /2. In articular, if we write ψ C (x, ) for the sum ψ C(x, ) with the m 2 terms removed, then (6.9) also holds for ψ C (x, ). Denoting the error term by R(x), and integrating by arts, π C (x, ) = (6.2) Since x 2 = x dψ C (n, ) 2 log n #C x # Gal() log x + R(x) log x #C # Gal() #C = # Gal() ( = O n /2 log n (log d L n n L ) x dn x 2 log 2 n R(n)dn 2 n log 2 n x ( ) dt 2 log t + O #C x # Gal() x/2 log(d L x n L R(n)dn ) 2 n log 2 n. ) #C # Gal() x/2 log(d L x n L ), (6.2) imlies Theorem 2.3. A similar argument (dealing with the more comlicated zero-free region) can be used to rove Alications of the Chebotarev Density Theorem To conclude, we give several interesting alications of the Chebotarev density theorem, chosen to highlight some of the different ways this theorem can be used. Our first examle demonstrates erhas the most obvious alication of Chebotarev showing that a certain set of rimes is infinite. This examle, described by Serre in Ser03] uses Chebotarev to translate a grou-theoretic theorem of Jordan into a number theory statement. Theorem 7.. Let f be an irreducible degree n 2 olynomial with coefficients in Z. Let P 0 (f) be the set of rimes such that f has no zeros in the finite field F. P 0 (f) has ositive density with strict inequality when n is not a rime ower. n Proof. We require some grou theoretic inut, which we state without roof. Theorem 7.2 (Jordan Jor72], Cameron and Cohen CC92]). Let G be a grou acting transitively on a set X with n 2 elements. Let G 0 = {g G : g x x x X}. Then, G 0. More recisely, #G 0, with equality ossible only when #G is a #G n ower of a rime. Let L = Qα,..., α n ], where α,..., α n are the roots of f Q. L is Galois over Q. Our strategy is to aly this result and Chebotarev with G = Gal(L/Q) and for C ranging over all conjugacy classes contained in G 0. Note that Gal(L/Q) acts transitively on the n-element set of roots of f in L. For any rime which is unramified in L and does not divide the leading coefficient of f, the degrees of the olynomials in the factorization of f over F give the cycle tye of

21 THE CHEBOTAREV DENSITY THEOREM 2 ] σ = L/Q viewed as an element of S n via the action on the n roots. Since the number of fixed oints of a ermutation is invariant under conjugation, the set G 0 Gal(L/Q) of fixed-oint free ermutations is a (disjoint) union of conjugacy classes in Gal(L/Q). So, (7.) { P 0 (f) = : L/Q ] G 0 }, u to a finite set of rimes. Alying the Chebotarev density theorem to the conjugacy classes C G 0, together with Cameron and Cohen s refinement of Jordan s theorem, we see that P 0 (f) has density /n (with strict inequality if n is not a ower of ). We briefly remark that this alication did not actually require the full ower of the Chebotarev density theorem, but follows from an easier theorem of Frobenius dating back to 880. In our notation above, Frobenius s theorem states that the density of rimes for which f factors as a roduct of degree d, d 2,..., d t olynomials is roortional to the number of elements of Gal(L/Q) which at on {α,..., α n } with cycle tye d, d 2,..., d t. See SL96] for a historical context and numerical examles exhibiting Frobenius s theorem. Another alication of this flavor shows that the set of rimes (in Z) reresented by the rimitive ositive definite quadratic form ax 2 + bxy + cy 2 is ositive and can be comuted in terms of the discriminant and symmetries of the form. The roof alies the Chebotarev density theorem for a certain dihedral Galois extension deending on the quadratic form. For details, see Theorem 9.2. of Cox89]. The Chebotarev density theorem can also hel to rove the infinitude of some set of rimes even when the set of rimes cannot be shown to have ositive density. As one ste in Elkies s roof that every ellitic curve over Q has infinitely many suersingular rimes Elk87], Elkies used Dirichlet s theorem to find an additional suersingular rime outside of any finite set of suersingular rimes. It seems ossible that a similar argument using the full ower of Chebotarev could rove the infinitude of some other set of rimes. As our next alication, we show how the Chebotarev density theorem can be used to show that two ieces of data determine each other. The idea is to use Chebotarev to roduce an infinite set of rimes and then to use a local-to-global theorem to say that the behavior of some data at an infinite set of rimes determines the data globally. The rototyical examle of such a result is the fact that x Q satisfies x = 0 if and only if x 0 (mod ) for infinitely many rimes. We closely follow the treatment in Cox89], although we strengthen the statement of the theorem. The result, which says that a Galois extension is determined by the rimes that slit comletely, is imortant for the study of ray class fields. Theorem 7.3 (8.9 of Cox89]). Let K be a number field and let L and M be Galois extensions of K. Let P := { O K rime : slits comletely in L}. Then, L = M if and only if there exist density zero sets of rimes Σ and Σ 2 such that P Σ = P Σ 2.

22 22 NICHOLAS GEORGE TRIANTAFILLOU Proof. By interchanging the roles of L and M, it suffices to rove L M if and only if there exists a density zero Σ such that P M/K P Σ. If L M, this is clear. Suose P M/K P Σ. Let N be the normal closure of L M over K, so N/K is Galois and contains L and M. Galois theory says that it is enough to show that any σ Gal(N/M) fixes L. Given σ Gal(N/M) Gal(N/K), by the] Chebotarev density theorem ] there are infinitely many rimes of K with σ N/K. Since σ fixes M, M/K =, whence P M P M slits comletely over. In articular, P M/K. Then, for infinitely many rimes, P. Now, α Nm = σ(α) (mod P) for all α N imlies that α Nm ] ] = σ L (α) (mod P L) for all α L, so σ L = L = = L since P. N/K P P L We could increase the density hyothesis to a small ositive number given a bound on the degree over K of the Galois closure of the comositum of L and M in terms of the extensions and M/K. For instance, the result still holds if we relace the hyothesis that Σ and Σ 2 have density zero by the hyothesis that Σ and Σ 2 have density strictly less than (L:K] M:K])! since the relative degree of the comositum is at most the roduct of the relative degrees of the base fields and the relative degree of the Galois closure is at most the factorial of the relative degree of the original extension. Our third alication relates to the comutational roblem of determining the Galois grou of, or at least its cycle structure. By comuting the Artin symbol at various rimes, the Chebotarev density theorem says that we will eventually find an element of every conjugacy class. The effective versions of the Chebotarev density theorem give a bound which rovides a guarantee that we will find a reresentative of each class after some finite comutation of known length. Moreover, the effective bounds, together with a bound of the size of the Galois grou can allow us to exactly determine the size of the grou and the conjugacy classes. This still isn t enough to in down the grou, but is a good lace to start. See the introduction to LO77] for a bit more discussion. Serre rovided another imortant class of alications of the Chebotarev density theorem in Ser8]. Even though the Chebotarev density theorem gives a result that certain sets of rimes have ositive density, it can be used to give better uer bounds on the sizes of certain sets of density zero rimes, articularly when dealing with infinite Galois grous. Assuming I understand the French correctly, in section 4 of Ser8], for G a comact l-adic Lie grou of M-dimension N, C a closed subset of G with M- dimension d that is stable under conjugation, and a reresentation ρ : Gal(E/K) G of an infinite Galois extension, Serre defines the counting function π C (x) to be the number of rime ideals O K of norm less than or equal to x such that is unramified in E and ρ ( E/K ]) (once it is aroriately defined for an infinite Galois extension) lies in C. Serre shows that π C (x) satisfies x π C (x) = O (ε(x) dn dx (7.2) log x (7.3) ε(x) = 2 { log x(log log x) 2 (log log log x), x /2 (log x) 2, ), as x, where we can take for x > 6 unconditionally, under GRH.