Math 620  Problem Set #6  Due Mon., Oct. 22, 2001

1. A function f on ℕ is called multiplicative if f(ab) = f(a)f(b) provided (a, b) = 1. a) Is ϕ multiplicative? Is μ? Is τ? (Here τ(n) = #{d ∈ ℕ | d|n}.) Is σ? (Here

σ(n) = ∑_{d|n} d.) Find explicit formulas for these functions evaluated at prime powers.

b) Suppose that f and g are related by g(n) = ∑_{d|n} f(d). Show that f is multiplicative

if and only if g is.

2. Find all solutions (x, y) to a) (5 + \sqrt{2})x + (2 − 3\sqrt{2})y = 1 in ℤ[\sqrt{2}].

b) (t^{2} + 1)x + (t − 1)y = 1 in ℤ/3 [t].

3. a) How many units are there in ℤ[i]/(6 + i)? Discuss the structure of the group of units. Do the same for ℤ[i]/(5).

b) Find a primitive root for U_{27} and use it to make a table of indices. (Hint: Work efficiently.) Use the table to find all solutions to the congruence x^{5} ≡ 22 (mod 27).

4. a) Show that ℤ[\sqrt{3}] has a division algorithm with respect to |N|, where N is a multiplicative integer-valued norm N such N(α) = 0 ⇔ α = 0. Namely, given α, β ∈ ℤ[\sqrt{3}] with β ≠ 0, there exist q, r ∈ ℤ[\sqrt{3}] satisfying α = βq + r and |N(r)| < |N(β)|. Can you improve the inequality?

b) What if instead of ℤ[\sqrt{3}], you consider the ring ℤ[(1 + \sqrt{3})/2]? the ring ℤ[\sqrt{5}]? the ring ℤ[(1 + \sqrt{5})/2]? (Note that to have a division algorithm, the norm must be integer valued.)

5. a) Find the continued fraction convergents to the numbers α in Problem Set 5, #4(a), which first approximate α to within 10^{−4}. Compare with Problem Set 5, #2(f).

b) Relate the convergents of \frac{1+\sqrt{5}}{2} to the Fibonacci numbers.

c) Compute p^{2} − 2q^{2} for each convergent p/q to \sqrt{2}. Similarly for \sqrt{3}, \sqrt{5}, \sqrt{7}. Conjecture?

6. a) For which integers m > 0 is there a solution to x^{2} + 2 ≡ 0 (mod m)?

b) Which primes in ℤ remain prime in ℤ[\sqrt{−2}]? Which split? Which ramify?

7. Let A = ℝ[t], B = ℝ[\sqrt{t}], A′ = ℂ[t], B′ = ℂ[\sqrt{t}].

a) Describe the primes of A. Which primes in A remain prime in B? Which split? Which ramify?

b) Redo part (a) with (A, B) replaced by (A′, B′).

c) Redo part (a) with (A, B) replaced by (A, A′).

8. Which of the following congruences have solutions?

(i) x^{2} ≡ 333 (mod 4001) (ii) x^{2} ≡ 333 (mod 4003) (iii) x^{2} ≡ 333 (mod 4005).

(Note: 4001 and 4003 are prime.)