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Math 515, Number Theory II, Spring 2000

**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.