Math 620  Problem Set #4  Due Fri., Oct. 5, 2001

1. a) Show that the ring of p-adic integers ℤ_{p} is a local ring with maximal ideal (p), and that the group of units in ℤ_{p} is the complement of (p).

b) Show that ℚ_{p} =^{def} ℤ_{p}[1/p] is a field, and is the fraction field of ℤ_{p}.

c) Show that ℤ_{p} is integrally closed (in its fraction field).

d) For which n, p is there a primitive nth root of unity in ℤ_{p}?

2. a) Let P be a polygon in the plane ℝ^{2}, such that every vertex is at a lattice point (i.e. a point with integer coordinates). Assume that the interior of P is simply connected but not necessarily convex. (Thus we rule out a large square with a small interior enclosed square removed, and we also rule out the union of two squares that touch just at one vertex. But we allow a large square with a small square along an edge removed.) Find a formula for the area of P in terms of the number of lattice points in the interior of P, the number in the interior of edges, and the number at vertices. [Hint: Try some examples.]

b) Let (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ ℝ^{2} be points in the first quadrant. Let P be the parallelogram whose vertices are at these two points, at the origin, and at (a_{1} + a_{2}, b_{1} + b_{2}). Show that

the area of P is the absolute value of the determinant of ( \begin{matrix}a_{1} & a_{2} \\ b_{2} & b_{2} \end{matrix} ), and interpret the sign

of the determinant.

c) Use the considerations in (a) and (b) to show that if \frac{a}{b}, \frac{c}{d} are consecutive Farey

fractions in the sequence F_{n} (see PS1 #5), then det ( \begin{matrix}a & c \\ b & d \end{matrix} ) = ±1, and hence \frac{c}{d} = \frac{a + e}{b + f}

if \frac{a}{b}, \frac{c}{d}, \frac{e}{f} are consecutive in some Farey sequence.

3. Let ϕ be Euler’s function. Prove that if n ≥ 1, then ∑_{d|n} ϕ(d) = n. (Hint: In the set

{1/n, 2/n, …, n/n}, how many elements, when reduced to lowest terms, have denominator equal to d?)

4. Show that there is a unique formal power series f(t) = a_{1}t + a_{2}t^{2} + ⋯ (with a_{i} ∈ ℝ) such that 3f(t)^{2} + f(t)e^{t} + sin t = 0. Find the first few coefficients a_{i}.

5. a) Prove the following strong form of Hensel’s Lemma: Let f(x) ∈ ℤ_{p}[x], x_{0} ∈ ℤ, m ≥ 0. Suppose f(x_{0}) ≡ 0 (mod p^{2m+1}) but f′(x_{0}) ≢ 0 (mod p^{m+1}). Then there is an α ∈ ℤ_{p} such that f(α) = 0 and α ≡ x_{0} (mod p). (Hint: Proceed as before, using Newton’s method, just as in the case m = 0.]

b) Deduce that there is a solution to x^{2} + 7 = 0 in ℤ_{2}, and hence a solution to x^{2} + 7 ≡ 0 (mod 2^{n}) for every n.

6. Use tables of indices to solve each of the following:

(a) x^{2} ≡ −1 (mod 97); (b) x^{10} ≡ 34 (mod 89); (c) x^{5} ≡ 52 (mod 61).