Math 621  Problem Set #7  Due Fri., April 19, 2002

1. Consider the extension L ⊃ K where K = ℚ and L = ℚ(i, \sqrt{5}).

a) Explicitly find the conductor c, the norm group N (c), and the Artin map σ : I(c)/P_{c}N (c) → Gal(L/K).

b) Verify Artin reciprocity in this example (i.e. that σ is an isomorphism).

c) What does Tchebotarev Density say for this example?

2. Verify the Existence Theorem for K = ℚ and c = 8v_{∞}. That is, for each subgroup H of I(c) containing P_{c}, explicitly find the abelian extension L of K that is unramified away from c and satisfies H = P_{c}N (c). In each case, verify that Artin Reciprocity holds.

3. Let K = ℚ(\sqrt{d}), where d is a square-free integer.

a) Show that if d is divisible by at least three primes, then K has an abelian unramified extension of degree 2. [Hint: Try K(\sqrt{d′}) for appropriate d′|d.]

b) In (a), show that the class number of K is even.

c) What if instead, d is divisible by fewer than three primes?

4. Let 1 → A_{1} → A_{2} → ⋯ → A_{n} → 1 be an exact sequence of finite groups, and let a_{i} = #A_{i}. Show that ∏_{i=1}^{n} a_{i}^{(−1)^{i}} = 1. [Hint: Reduce to the case n = 3.]

5. Let a, q, r > 1 be integers, with q prime.

a) Let g(x) = (x^{q} − 1)/(x − 1) ∈ ℤ[x], and let α = a^{q^{r−1}}. Show that α − 1 and g(α) are each greater than 1.

b) Show that if p is a prime with p|g(α) and p|(α − 1), then p = q. [Hint: Show g(α) ≡ q (mod p).]

c) Show that if q|g(α) and q|(α − 1), then g(α) is not a power of q. [Hint: Write α = β + 1 and expand g(α). Can q^{2}|g(α)?]

d) Deduce that if q|g(α) and q|(α − 1), then there is a prime p ≠ q that divides g(α). Show that p does not divide α − 1. [Hint: Parts (b),(c).]

e) Show that if q is prime to g(α) or to α − 1, and if p is any prime dividing g(α), then p does not divide α − 1. [Hint: Part (b).]

f) Deduce that there is a prime p that divides g(α) but not α − 1. Show that p divides a^{q^{r}} − 1 but not a^{q^{r−1}} − 1.

g) Conclude that the order of a ∈ (ℤ/p)^{∗} is q^{r}, for some prime p.

h) Does part (g) still hold if q^{r} is replaced by an integer that is not a power of a prime?

6. Let n, a, N be positive integers. Show that there exist positive integers m, b such that a, b are prime to m; the subgroups ⟨a⟩, ⟨b⟩ ⊂ (ℤ/m)^{∗} intersect trivially; m is relatively prime to N!; and n divides the orders of a and of b in (ℤ/m)^{∗}. [Hint: Consider the prime factorization n = q_{1}^{e_{1}} ⋯ q_{s}^{e_{s}}. Pick r_{i} > max(2, e_{i}, N) and apply Problem 5(g), obtaining a prime p_{i}. Let p = max(p_{1}, …, p_{s}) and choose r′_{i} > max(r_{i}, p). Apply Problem 5(g) again and get a prime p′_{i} > p. Let m = (p_{1}p′_{1}) ⋯ (p_{s}p′_{s}). Now use the Chinese Remainder Theorem to find b > 1 such that b ≡ a (mod p_{1} ⋯ p_{s}) and b ≡ 1 (mod p′_{1} ⋯ p′_{s}). Verify that these choices work.]