D. J. Bernstein
More mathematics
Math 515, Number Theory II, Spring 2000

What I did in class

20000110: Course information. Prerequisites. Diophantine motivation for fractions, algebraic extensions, completions. Building a typical local/global diagram in the number field case: Z, Q, Z[sqrt(-5)], Q[sqrt(-5)], completions at 5. Algebraic completions versus topological completions; maximal ideals versus valuations; warning about completions at infinity. Typical function field example: F_7[sqrt(x)] etc. Notes on the analogy between number fields and function fields. Course contents: Class field theory. Goal of classifying abelian extensions of number fields. Easy special case of Cebotarev's density theorem: distribution of primes mod 10. Course outline: define Frobenius symbol; state density theorem; prove density theorem. Textbook: Janusz, second edition. Reference books: Cassels-Froehlich; Koch.

20000112: Examples of finite fields: F_2; F_8 as F_2[x]/(x^3+x+1). The Galois group of F_8 over F_2. Classification of finite domains from first principles. The Frobenius automorphism. Classification of automorphisms of a finite field. Classification of ring homomorphisms from one finite field to another.

20000114: Setup: Galois extension L:K of number fields, Galois group G; maximal ideal Q of O_L over maximal ideal P of O_K; extension l:k of residue class fields. Elements of G take O_L to O_L, primes over P to primes over P. Transitivity on primes over P. Equality of e's and f's. #G = e(Q:P) f(Q:P) #{primes over P}. The decomposition group G_Q. The inertia group T_Q; higher ramification groups V_{1,Q} et al. The natural homomorphism from G_Q to Gal_k l. T_Q as the kernel of the homomorphism. #G_Q = e(Q:P) f(Q:P) by transitivity. Consequences of #(G_Q/T_Q) >= f(Q:P): #T_Q = e(Q:P); #(G_Q/T_Q) = f(Q:P); the homomorphism is surjective. When Q is unramified: G_Q isomorphic to Gal_k l; definition of the Frobenius symbol of Q in L:K. Concrete characterization of the Frobenius symbol. Frobenius symbols of all primes over P forming a conjugacy class. What Cebotarev's density theorem says.

20000119: Picture of L, the inertia field, the decomposition field, K; all the primes under Q; the residue class fields. f(Q:decomposition field) >= f(Q:P) using multiplicativity of e's. f(Q:inertia field) = 1 using the inertia polynomial trick. Conclusion: #(G_Q/T_Q) >= f(Q:P). Picture of e's and f's under Q: all the e on top, all the f in the middle. Comments on the names ``decomposition'' and ``inertia.'' The local case. Extending an element of G_Q to an automorphism of L_Q over K_P. Compare degrees: L_Q over K_P is Galois with group isomorphic to G_Q.

20000121: Example of a Galois extension: K = rationals, L = K[x]/(x^2-x-3), ring of integers = Z[x]/(x^2-x-3), Galois group G = {1,conj} where conj x = 1-x. Constructing a prime ideal over 3 from Z[x] -> (Z/3)[x]/(x-1). Another prime ideal over 3; e's and f's. Factorization of 3; extensions of valuation. Galois action on primes over 3: trivial decomposition group, trivial Frobenius. Constructing the unique prime ideal over 2. Nontrivial decomposition group, nontrivial Frobenius. Kummer's criterion for primes of Z[x]/phi when phi is a monic irreducible polynomial.

20000124: Factorization of 13; ramification; extension of valuation. Factorization of generic primes where 13 is a square. Factorization of generic primes where 13 is not a square. Summary: Frobenius symbol generalizes Legendre symbol. Another example of a Galois extension: sqrt(-1), sqrt(2), sqrt(5). Analysis of how the prime 5 factors. Decomposition field and inertia field.

20000126: Order of the Frobenius symbol. Special cases: splitting completely, remaining inert. How the Frobenius symbol responds to conjugation. Notation in the abelian case. Frobenius symbol over an intermediate field. Frobenius symbol under an intermediate field; the restriction map. The 13th cyclotomic field. Galois group of the 13th cyclotomic field. The Frobenius symbol of a prime other than 13. Order of the Frobenius symbol. Fact that the 13th cyclotomic field contains sqrt(13). Quadratic reciprocity via the Frobenius symbol.

20000128: Facts about the 13th cyclotomic field: irreducibility of the 13th cyclotomic polynomial; ring of integers; ramification only at 13. Constructing 12 different automorphisms of the field. Comparing degrees to show that the field is Galois over the rationals. The normal subgroup of squares inside the Galois group; the fixed field. Picture of fields with degrees; the restriction map. Proof using ramification that the fixed field is sqrt(13). Alternate proof using a Gauss sum. Restriction of Frobenius; quadratic reciprocity. How 3Z splits. How 5Z splits. Relating Cebotarev's density theorem for the 13th cyclotomic field to the distribution of primes mod 13.

20000131: Definition of natural density. Definition of Dirichlet density. Relating the zeta function to Dirichlet density. Setup for using Frobenius to understand an arbitrary extension M:K of number fields: number field L; subgroups H < G < Aut L; fields M = L^H and K = L^G; maximal ideal Q of O_L, unramified over P of O_K; Frobenius symbol s = Frob_{L:K} Q. How s permutes the right cosets of H in G. Correspondence between cycles of s and primes of M over P; cycle lengths are f's; explicit correspondence. Proof that each element of a cycle gives the same prime. Proof that different cycles give different primes. Proof that f's are at least cycle lengths. Adding up ef's to show that f's equal cycle lengths and that there are no other primes.

20000202: The splitting field of x^3-2. The Galois group as permutations of the roots. Explicit elements of the Galois group of orders 2 and 3. The subgroup generated by complex conjugation. The fixed field: Q[cbrt(2)]. Picture of Q, Q[cbrt(2)], splitting field. Factoring x^3-2 mod 5 to find primes over 5 with f=1 and f=2. Filling in the picture of primes; relating e's and f's to explicit Frobenius cycle structure. Setup for the Artin map: abelian extension L:K of number fields. Extending Frob to products of powers of unramified primes.

20000204: The Artin map. The Artin map for the 13th cyclotomic field. The Artin map for sqrt(13). The Artin map over an intermediate field; norms of ideals. Statement of how the Artin map behaves in a translated field extension. The lemma behind weak approximation.

20000207: Weak approximation. Example comparing weak approximation to the Chinese Remainder Theorem. Definition of moduli. Definition of congruence mod* a modulus. Definition of the ray K_{m,1}. Definition of the group I_K^m. Definition of iota. Note that iota K_{m,1} is contained in I_K^m. Example: K_{1,1}, I_K^1, usual class group. Example: K_{1,P^n}, I_K^{P^n}, ray class group mod P^n. Example: ray class group of the rationals at infinity. How K_{m,1} and I_K^m react to multiplication of coprime moduli. Definition of K_m. Fact that K_m/K_{m,1} is multiplicative.

20000209: Example: ray class group of the rationals at 2^2 infinity; comparison to Frobenius symbol of Q(sqrt(-1)). Proof that K_m/K_{m,1} is multiplicative. K_m/K_{m,1} is finite. Proof that all ray class groups are finite. Proof that I_K^m/principal is isomorphic to the usual class group; using the Chinese remainder theorem to avoid primes. The Riemann zeta function. The zeta function of a ray class.

20000211: Uniform convergence and analyticity of sum_n (f_n-f_{n-1})/n^s. Picture of convergence region. Proof. Example: f_n = n, Riemann zeta function. Example: f_n = n mod 2; value at 1; comparison to Riemann zeta function; analytic continuation of Riemann zeta function around 1; residue of Riemann zeta function at 1. General statement of residue at 1 when f_n/n converges. Example: f_n = #{ideals of O_K in C of norm at most n} when C is a ray class.

20000214: How the Riemann zeta function implies existence of infinitely many primes. The zeta functions for the rationals mod 2^2 infinity: behavior around s=1; L-series; factorization of L-series; conclusions about primes congruent to 1 mod 4 and 3 mod 4.

20000216: Recap of proof of Dirichlet's theorem mod 4. The zeta function for sqrt(-1). Unique generators for ideals of Z[sqrt(-1)]. Counting generators of norm below n. Factorization of the zeta function. What happens for f=1; what happens for f=2. Conclusion that there are infinitely many primes congruent to 1 mod 4. Comparison of the zeta function to the product of the L-series in the previous example. The zeta function for sqrt(2). Dirichlet's Log map for sqrt(2). Unique generators, in terms of Log, for ideals of Z[sqrt(2)].

20000218: Unique generators, not in terms of Log, for ideals of Z[sqrt(2)]. Picture. Picture of generators of norm at most n. Estimating the number of ideals in terms of the volume. Notes on computing the volume. Infinitely many degree-1 primes in Z[sqrt(2)]. Structure of units near 1 mod m.

20000221: Setup for the analytic class-number formula: number field K, real and complex embeddings; roots of 1, the log map, the vector W = (1,...,2), Dirichlet's unit theorem; a modulus m, Dirichlet's unit theorem mod m; fundamental units u_1,u_2,...,u_{r+s-1} mod m; a fundamental region in log-space. The region Gamma_x: vectors (y_1,...,y_{r+s}) with log in the fundamental region, sum of log coordinates at most x, positive coordinates at the infinite places in m. How Gamma_x scales with x. Further setup, given a ray class C mod m: finding an ideal D in the inverse of C; finding alpha_0 in D intersect K_{m,1}; the set A_n = everything in alpha_0+m_0 D with embeddings in Gamma_x where x = log(n Norm D). The determinant of the lattice alpha_0+m_0 D. Approximating #A_n in terms of the volume of Gamma_0. Conclusion that #A_n/n has limit independent of C. Restatement of the definition of A_n; restatement of the fundamental region. A function on A_n. Proof that the function produces ideals in C of norm at most n. Proof that the function is surjective.

20000223: Proof that the function is w-to-1 where w is the number of roots of 1 in K_{m,1}. Approximating f_n/n in terms of the volume of Gamma_0, where f_n is the number of ideals in C of norm at most n. Definition of B-Lipschitz functions. Statement of a general theorem on lattice points. The error in the f_n/n approximation. Restatement of the definition of Gamma_0. Switching to polar coordinates: the region X; the volume of Gamma_0 as an integral over X. Substituting (log rho_1,...,2 log rho_{r+s}) = (log N)W/(r+2s) + sum_j c_j Log u_j. (In retrospect, I should have defined W as (1,...,2)/(r+2s).) Image of X under the substitution: the unit cube. Sum of log rho equalling log N. Derivative of the substitution with respect to N; derivative of the substitution with respect to c_j. The regulator of K mod m and the Jacobian of the substitution. The volume of Gamma_0 in terms of the regulator. Fact that the volume is nonzero. Conclusion: the limit of f_n/n in terms of the regulator; error in the approximation.

20000225: The analytic class-number formula. The analytic class-number formula for modulus 1. What the analytic class-number formula has to do with class numbers: add over all classes. Example: sqrt(2) again; embeddings, fractional ideals, roots of 1, discriminant, ring of integers, units, regulator; approximate formula for number of nonzero ideals of norm at most n. Analytic continuation around 1 of the zeta function for C. Behavior at 1. Adding over all ideal classes: how the zeta function for K behaves at 1. Comment on the Birch-Swinnerton-Dyer conjecture. Characters of the ray class group. The L-series for a character. The L-series for the trivial (``principal'') character. Behavior of the L-series near 1 for nontrivial characters. Product expansion of the L-series.

20000228: Absolute convergence of the product. Convergence to the L-series. All characters of a finite commutative group. What happens when L-series are multiplied over all characters.

20000301: Setup for applications of Dirichlet density: number field K, modulus m; zeta functions for ray classes; L-series for characters of the ray class group; product expansion; subgroup H of I_K^m containing iota K_{m,1}. Definition of Dirichlet density of T as lim_{s->1+} (log prod_{P in T} 1/(1-Norm(P)^(-s)))/(log(1/(s-1))). Comments on other definitions of density: removing the reciprocals; approximating the logarithm of 1/(1-Norm(P)^(-s)); requiring stronger convergence to the limit. Proof that the density of the set of all maximal ideals is 1. Proof that any density that exists is between 0 and 1. Case 1, some L-series has limit 0 at 1: density of primes in H is 0. Case 2, no L-series has limit 0 at 1: density of primes in any coset of H is 1/#(I_K^m/H), i.e., primes are uniformly distributed among the cosets, as measured by density.

20000303: Setup for the Frobenius density theorem (abelian case): extension L:K of number fields, commutative Galois group G. First form of the theorem: for any subgroup H of G, the density of primes with Frobenius symbol in H is #H/#G. Proof. Additivity of the density=#S/#G property over subsets S of G; comments on the group of multisubsets of G. Divisions of G: X_sigma = {g in G:<g> = <sigma>}. Second form of the theorem: for any sigma, the density of primes with Frobenius symbol in X_sigma is #X_sigma/#G. Proof. Surjectivity of the Artin map from I_K^m to G, when it is defined. The first fundamental inequality of class field theory: for H=(Norm_{L:K} I_L^m) iota K_{m,1}, the density of primes in H is at least 1/#G, so #(I_K^m/H) is at most #G; truth of case 2; primes are uniformly distributed among cosets of H.

20000306: Extended example: K = rationals, m = 5 infinity, H = iota K_{m,1}. Descriptions of I_K^m, K_{m,1}, H. Cosets of H in I_K^m. List of ideals in H, 2H, 3H, 4H. The zeta function of H. Comparison to the Riemann zeta function. Definition of g_1. Convergence of the Riemann zeta function for s > 1; limit of zeta(s)-1/(s-1) exists. Convergence of the zeta function of H for s > 1; convergence of g_1 for s > 0; conclusion that the limit of zeta_K(s,H)-1/5(s-1) exists. Similarly for 2H, 3H, 4H. Characters of I_K^m/H as powers of one character chi. Table of values of characters. L(s,1) as sum; L(s,1) as product; limit of L(s,1)-4/5(s-1) exists. L(s,chi) as sum; L(s,chi) as product; limit of L(s,chi) exists. Similarly for L(s,chi^2), L(s,chi^3). Product of all L-series as a product over primes.

20000308: Case 1. What case 1 would mean for density of primes in H. Note on easy ad-hoc way to see in this example that case 1 is false. Case 2. Density of primes in H. Product of appropriate powers of L-series to compute density of primes in 2H. The fifth cyclotomic field. Frobenius symbols; ramification. The analytic class-number formula for the field mod 5. The zeta function for the field mod 5 as the product of the L-series for K mod 5; so case 2 is true.

20000310: Chapter 5 setup: extension L:K of number fields, commutative Galois group G, ``big enough'' modulus m for K, H = (Norm_{L:K} I_L^m) iota K_{m,1}. Statement of the Artin reciprocity theorem: the Artin map from I_K^m onto G has kernel H. Statement of the reciprocity law: the kernel of the Artin map contains iota K_{m,1}. Statement of the fundamental equality of class field theory: #(I_K^m/H) = #G. Statement of Cebotarev's density theorem (abelian case): the density of primes with Frobenius symbol sigma is 1/#G. Strategy: assume G cyclic; count sizes of various groups; #(I_K^m/H) is a multiple of #G, so it equals #G; prove the reciprocity law; H equals the kernel of the Artin map; finally Cebotarev. How to immediately see truth of case 2, given the reciprocity law: zeta function is a product of L-series. Statement of the existence theorem. Correspondence between abelian extensions of K (in the complex numbers) and pairs (m,H) mod an equivalence relation. Definition of g-module. Definition of Delta, N, H^0, H^1, Herbrand quotient.

20000313: No class.

20000315: No class.

20000317: No class.

20000320: Recap of definition of Herbrand quotient. Example: Z with the identity action has Herbrand quotient 1/g. Herbrand quotient of a finite g-module is 1. Herbrand quotient multiplies over exact sequences; proof using the long exact sequence from homology theory. Brief comments on the Grothendieck group. Cyclic extension L:K of number fields, Galois group generated by sigma, order g; g-module structure on the group of fractional ideals generated by all primes of L over P. Herbrand quotient of this group is 1/ef. (Who decided to use #H^1/#H^0 instead of #H^0/#H^1? Next time I'll take the reciprocal.)

20000322: Modulus m for K, divisible by P; g-module map from the group of fractional ideals generated by all primes of L dividing m to the group of fractional ideals generated by all primes of L over P. Kernel of the map. Conclusion: Herbrand quotient is product over finite P of 1/ef. Herbrand quotient of principal ideals in the group is the same, because Herbrand quotient of class group is 1. First step in computing Herbrand quotient of units: constructing Dirichlet-type units permuted by the Galois group. The g-module sum_v sum_{w|v} Zw; a g-module map to units.

20000324: Example: Q[sqrt(2)] over Q; mapping (x,y) to (1+sqrt(2))^x (1-sqrt(2))^y; kernel generated by (2,2). Constructing a nonzero element of the kernel in general: local norms of the units are dependent. This element is fixed by the g-module action. Rank of the lattice of logarithms of the image of the map. Rank of the kernel. Kernel is isomorphic to Z as a group. The g-module structure must be trivial since it fixes a nonzero element. Herbrand quotient of the kernel. Herbrand quotient of sum_v sum_{w|v} Zw. Herbrand quotient of the image. Herbrand quotient of units is g/product e(v). Definition of m-units. Herbrand quotient of m-units. Introduction to Q-adic exponentials.

20000327: Setup for local computations: cyclic extension L:K of number fields, extension Q:P of finite primes, over 2 for simplicity; completions at Q; local Galois group generated by sigma, of order g. (Note that these are generally not the same as the global sigma and g.) Some g-modules: L_Q and L_Q^*. Some sub-g-modules. The Herbrand quotient of O_Q under addition is 1, using fact that O_P-submodule of O_Q generated by a normal K_P-basis of L_Q has finite index. Definition of the Q-adic exponential. Convergence. Outline of proof that exp is a bijection from Q_Q^n under addition to 1+Q_Q^n under multiplication, when n is large enough. Proof that exp is a group map. Proof that exp is a g-module map. Herbrand quotient of 1+Q_Q^n. Herbrand quotient of O_Q^*.

20000329: Low-tech proof that the index was finite. Size of H^1(O_Q^*) is e: L_Q^*/K_P^*O_Q^* as the kernel of two maps. Conclusion that norms of O_Q^* have index e in O_P^*. Everything in 1+P_P^n is a norm from O_Q^*, if n is large enough. Translating local to global: (K_m intersect Norm_{L:K} L^*)K_{m,1} has index e in K_m. Proof that the map from K_m to O_P^*/norms is surjective, kernel contains the alleged kernel; writing global norms as local norms.

20000331: Proof that the alleged kernel contains the kernel. Back to the global situation. Index of (Norm_{L:K} L^*)K_{m,1} in K^* is e(P)f(P), if n is large enough and m = P^n. Analogous statement for real primes; outline of proof. Comment on f notation for infinite primes. Preview of multiplicativity of the index.

20000403: Extending a modulus from K to L. Proof that Norm_{L:K} L_{m,1} is contained in K_{m,1}. Proof that index of (Norm_{L:K} L^*)K_{m,1} in K^* is multiplicative. Conclusion: index is the product of e over infinite primes in m, times the product of ef over finite primes in m, if each finite prime appears to a large enough power. If phi is a g-module map from X to Y and H^1(X) = H^1(Y) = 1 then #Cok/#Ker for the induced map on H^0 equals the Herbrand quotient of Ker phi divided by the Herbrand quotient of Cok phi. Extracting this from Herbrand exactness when H^0 is finite.

20000405: Proof without finiteness assumption on H^0. Setup for the fundamental equality: cyclic extension L:K of number fields, Galois group generated by sigma, order g, m modulus for K, m divisible by all ramified infinite primes, m divisible by all ramified finite primes, all finite primes in m appear to large enough powers. Basic facts about L^* -> I_L^m: H^0, H^1, kernel is set of m-units, finite cokernel, the induced map on H^0 takes K_{m,1} to iota K_{m,1}. Combining Herbrand quotient of m-units with size of K^*/(Norm_{L:K} L^*) K_{m,1}. Conclusion: the fundamental equality. Proofs of the basic facts.

20000407: Picture of different formulations of reciprocity; using the fundamental equality to move within the picture. For which m is iota K_{m,1} contained in the Artin kernel? Answer: all multiples of one m, the conductor of L over K. Examples of reciprocity: quadratic reciprocity, cyclotomic reciprocity. Proof that all multiples of the conductor are admissible. How to use weak approximation to see that the gcd of two admissible m's is another one.

20000410: The nth cyclotomic extension of a number field K. The extension is Galois; each conjugate of zeta_n is a power of zeta_n; embedding the Galois group into (Z/n)^*. The product of (1-zeta_n^i) for i in {1,2,...,n-1}; Frob P by process of elimination, when P is unramified and does not divide n. Kummer's criterion again; Dedekind's criterion for singularity; if P does not divide n then P must be unramified since x^n-1 is squarefree mod P.

20000412: Rephrasing in terms of discriminants the proof that P is unramified if P does not divide n. (Even faster is to observe the lack of ramification as a side effect of eliminating all but one possibility for Frob P.) Fun with polynomials: explicit factorization of x^n-1 over F_q, without using Frob facts. Back to Frob: the reciprocity law for cyclotomic extensions. Artin's reciprocity theorem for cyclotomic extensions, using the first fundamental inequality. First step in reciprocity for cyclic extensions: constructing a prime for which a given number has a given odd prime order. Constructing a prime for which a given number has a given prime-power order larger than 2.

20000414: The theorem on natural irrationalities. Some applications. Constructing cyclotomic extensions for Artin's reciprocity theorem: if L:K is an extension of number fields, g and s are positive integers, and P is a finite prime of K unramified in L, then there exists a positive integer n and an element tau of the Galois group of K(zeta_n) over K such that n is coprime to s and P and disc L, g divides the order of the Frobenius symbol of P in K(zeta_n) over K, g divides the order of tau, and the group generated by tau has trivial intersection with the group generated by Frob P. L intersect K(zeta_n) equals K if n is coprime to disc L.

20000417: Artin's reciprocity theorem for sub-cyclotomic extensions. If L:K is a cyclic extension of number fields, m is a modulus for K divisible by all finite primes ramified in L, and J is in the Artin kernel mod m, then J is in Norm_{L:K} I_L^m times iota K_{m,1}. Proof (when J has two primes, for simplicity) by crossing L:K with (two) cyclotomic extensions.

20000419: A note on composites of number fields. The Frob proof of irreducibility of cyclotomic polynomials over Q. Artin's reciprocity theorem for cyclic extensions. Artin's reciprocity theorem for abelian extensions. Cebotarev's density theorem for abelian extensions.

20000421: Cebotarev's density theorem for Galois extensions. Typical application (already proved by Frobenius): density of primes with a given factorization type in L is the density of permutations with that cycle type in the Galois closure of L.

20000424: What the existence theorem says. Class fields; ray class fields; the Hilbert class field. Picture of class fields over Q for modulus 1; 1 infinity or 2 or 2 infinity or 3 or 4 or 6; 3 infinity or 6 infinity; 4 infinity; 5 infinity; 7 infinity; 8 infinity. Picture of class fields over Q(sqrt 10) for modulus 1; modulus 2 infinity. Galois groups of unramified extensions of Q(cbrt 113). Class field for H intersect H' is composite of class fields for H and H'; uniqueness of class fields; every class field is contained in a ray class field.

20000426: Every abelian extension is contained in a ray class field. The Kronecker-Weber theorem. Hilbert class field is the maximal unramified abelian extension. Primes are uniformly distributed in any ray class group. How to construct class fields? Cyclotomic extensions; notes on analogous construction for complex quadratic fields. Kummer extensions; this is enough to prove the existence theorem. Definition of adeles. Definition of ideles. Examples for Q. Topology on adeles. Topology on ideles.

20000428: Principal ideles. Ideles onto fractional ideals. Norms of ideles. The commutative diagram of norms. Ideles onto a ray class group. Ideles onto the Galois group of an abelian extension; kernel. A finite abelian extension of K in C corresponds to a closed subgroup of ideles containing principal ideles and having finite quotient. Summary of infinite Galois theory: the Krull topology on Aut L; the Galois group of the fixed field of S is the closed subgroup generated by S; the fundamental theorem. Summary of infinite class field theory: an abelian extension of K in C corresponds to a closed subgroup of ideles containing principal ideles and having totally disconnected quotient.