Number Theory Assignment

due Monday, May 1, 2000

1. Exhibit Eisenstein units ϵ_{i} and Eisenstein integers ξ, η, and ζ such that

ϵ_{1}ξ^{3} + ϵ_{2}η^{3} + ϵ_{3}ζ^{3} = 0 and ξηζ ≠ 0 .

2. (a) Show that any Eisenstein integer not congruent to 0 mod 2 is congruent to some power of ω mod 2, where ω = (1 + \sqrt{−3})/2.

(b) Show that any Eisenstein integer not congruent to 0 mod 5 is invertible as an Eisenstein integer mod 5.

(c) Exhibit an Eisenstein integer that is invertible mod 5 but that is not congruent mod 5 to an Eisenstein unit.

3. Let F_{p} denote the field of p elements when p is prime. Let M_{p} denote the number of points (x, y) in F_{p} × F_{p} lying on the “curve” y^{2} = x^{3} + 1 .

(a) For each prime p ≤ 50 determine the number M_{p}. Present your results in tabular form.

(b) Are you able to prove any general statements about M_{p}?

4. Over the field of rational numbers find the first 6 “multiples” of the point (2, 3) in the arithmetic on the elliptic curve y^{2} = x^{3} + 1 .

5. Find three pairwise non-similar right triangles with rational sides having area 6.

Document network location:

DVI: http://math.albany.edu:8000/math/pers/hammond/course/mat4525/assgt/nt000414.dvi
HTML: http://math.albany.edu:8000/math/pers/hammond/course/mat4525/assgt/nt000414.html