Math 620  Problem Set #7  Due Mon., Oct. 29, 2001

1. a) For which primes p is there a solution to x^{2} = 13 in ℤ/p? in ℤ_{p}? in ℚ_{p}? b) Do the same with 13 replaced by 19.

2. Let A = ℤ[x]/(x^{2} − 2), which is abstractly isomorphic to ℤ[\sqrt{2}]. Let ϕ_{1}, ϕ_{2} : A → ℝ be given by ϕ_{1}(a + bx) = a + b\sqrt{2}, ϕ_{2}(a + bx) = a − b\sqrt{2}. (Here \sqrt{2} denotes the positive square root to 2.)

a) Show that ϕ_{1}, ϕ_{2} are injective homomorphisms of rings, and that each has dense image in ℝ.

b) Show that if α ∈ A then either α = 0 or else |ϕ_{1}(α)ϕ_{2}(α)| ≥ 1.

c) Let ϕ : A → ℝ × ℝ be given by ϕ(α) = (ϕ_{1}(α), ϕ_{2}(α)). Show that the image ϕ(A) is a discrete subset of ℝ × ℝ, and find the minimum distance between distinct points of ϕ(A). [Hint: First use (b) to show that the origin is an isolated point in the image of ϕ.]

3. Let A be as in problem 2.

a) Show explicitly that there are infinitely many units in A. [Hint: The units form a group.]

b) Show that u ∈ A is a unit if and only if ϕ(u) ∈ ℝ × ℝ lies on one of the hyperbolas xy = ±1. (Here ϕ = (ϕ_{1}, ϕ_{2}) as in problem 2(c) above.)

c) Show that if u ∈ A is a unit and ϕ_{1}(u) > 1, then u is of the form a + bx with a, b > 0. [Hint: First show that if ϕ_{1}(u) > 1 then part (b) implies that |ϕ_{2}(u)| < |ϕ_{1}(u)|.)

d) Let U = ℤ[\sqrt{2}]^{∗}, the group of units in ℤ[\sqrt{2}]. Consider the subset S ⊂ ℝ given by S = \{u ∈ ℤ[\sqrt{2}] | u ∈ U, u > 1\}. Deduce that S has a least element δ, and then find δ explicitly. [Hint: Part (c).]

e) Show that every unit in ℤ[\sqrt{2}] is of the form ±δ^{n} for some n ∈ ℤ. Then conclude that the group of units U in ℤ[\sqrt{2}] is isomorphic to the group ℤ/2×ℤ via the isomorphism ℤ/2 × ℤ → U given by (j, n) ↦ (−1)^{j}δ^{n}.

4. a) Let P be the set {2, 3, 5, 7, 11, …, ∞} of prime numbers together with infinity. For p ≠ ∞, let | |_{p} denote the p-adic absolute value; let | |_{∞} denote the usual absolute value. Show that if a is a non-zero integer, then ∏_{p∈P} |a|_{p} = 1.

b) Prove an analog of this with ℤ replaced by ℤ[i]. (You’ll have to figure out what |a|_{3}, |a|_{2+i}, |a|_{1+i}, |a|_{∞}, etc. mean here.)

c) Prove an analog for ℤ[\sqrt{2}]. (Note that there are now two infinite primes, i.e. two inequivalent embeddings into ℂ.) Show that this gives another proof of #2(b) above.

d) Prove an analog for ℂ[x]. [Hint: What are the primes, and how can you interpret them geometrically? What is the “prime at infinity”, and what does it correspond to geometrically on the Riemann sphere? If 𝔭 is a prime and f ∈ ℂ[x], interpret |f|_{𝔭} in terms of the “order of vanishing” of f there.]

5. a) Suppose that p is a prime, that p ≡ 1(mod m), and that a ∈ (ℤ/p)^{∗}. Show that a is an mth power in ℤ/p if and only if a^{\frac{p−1}{m}} = 1 ∈ ℤ/p.

b) Show that 16 is an 8th power in ℤ/p for every p, even though it is not an 8th power in ℤ. [Hint: Consider p mod 8, and in each case show that either 2 or −2 is a square, or else that −4 is a fourth power. Part (a) may be helpful in the last case.]