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

# Course handouts

20000114: Exercise 1: Let L be a number field, G a subgroup of Aut L, Q a maximal ideal of O_L. Define G_Q = {g in G: gQ = Q}. Consider the map g |-> (z + Q |-> gz + Q) from G_Q to Aut(O_L/Q). Prove that this map is a group homomorphism.

20000119: Exercise 2: Under the hypotheses of Exercise 1, let g be an element of G_Q. Prove that there is a continuous function h: L_Q -> L_Q such that h on L = g, and prove that h is a ring homomorphism.

20000121: Exercise 3: Find all the prime ideals of Z[x]/(x^3-4x-1) over 2, 3, 5, 7.

20000128: Exercise 4: Textbook, page 134, #2.

20000207: Exercise 5: Find a rational number z with ord_2 (z-314) >= 10, ord_3 (z-159) >= 10, ord_5 (z-265) >= 10, and |z-358| <= 0.001.

Exercise 6: Let K be a number field, m a modulus for K. Prove that y/z is in K_{m,1} if y and z are both in K_{m,1}.

20000209: Exercise 7: Let y be a positive rational number with odd numerator and denominator. Prove that ord_2 (y-1) is at least 2 if and only if Frob_{sqrt{-1}} y = 1.

20000228: Exercise 8: Let K be a number field. Let n be a positive integer. Let s be a real number, at least 1. Prove that the product of 1+n(Norm P)^(-2s), over all maximal ideals P of O_K, is at most zeta(2)^(nd), where zeta is the Riemann zeta function and d is the degree of K.

20000301: Exercise 9: Let K be a number field. Let T be a set of maximal ideals of O_K, not containing any degree-one primes. Prove that the density of T is 0.

Exercise 10: Let K be a number field. Let T be a finite set of maximal ideals of O_K. Prove that the density of T is 0.

Exercise 11: Let K be a number field. Let T and T' be disjoint sets of maximal ideals of O_K, each having a density. Prove that the density of T union T' is the density of T plus the density of T'.

20000310: Exercise 12: Let L:K be an abelian extension of number fields. Let m be a modulus for K divisible by all finite ramified primes. Let H be the kernel of the Artin map on I_K^m. Assume that H contains iota K_{m,1}. Let s be a real number larger than 1. Consider characters of I_K^m/H. Prove that the sum of (Norm J)^(-s) over ideals J in I_L^m is the product over characters chi of the product over prime ideals P in I_K^m of 1/(1-chi(P)(Norm P)^(-s)).

20000327: Exercise 13: Let L:K be an abelian extension of number fields such that all principal fractional ideals are in the kernel of the Artin map. Prove that the class number of K is a multiple of the degree of L over K.

20000403: Exercise 14: Let A, B, C be commutative groups. Let f be a group homomorphism from A to B such that Ker f and B/fA are finite. Let g be a group homomorphism from B to C such that Ker g and C/gB are finite. Prove that #(C/gfA)/#Ker gf is the product of #(C/gB)/#Ker g and #(B/fA)/#Ker f.

20000405: Exercise 15: Let A, B be commutative groups. Let f be a group homomorphism from A to B such that Ker f and B/fA are finite. Let C be a subgroup of A such that A/C is finite. Prove that #(B/fC) is a multiple of #(A/C)#(B/fA)/#Ker f.