Number Theory Assignment

due Monday, March 27, 2000

1. Parameterizing the rational points on the unit circle:

(a) Verify that for each rational number t the point

R(t) = (\frac{t^{2} − 1}{t^{2} + 1}, \frac{2t}{t^{2} + 1})

is a point (with rational coordinates) on the unit circle.

(b) What is the limiting value of R(t) as t approaches ±∞?

(c) For a given rational number t find the point (x, y), other than (1, 0), on the unit circle where the line L_{t} through the points (1, 0) and (0, t) intersects the unit circle.

(d) Given a point (x, y), other than (1, 0), on the unit circle find the point on the vertical coordinate axis where the line through (1, 0) and (x, y) intersects the vertical axis.

(e) Show that each point on the unit circle with rational coordinates, other than (1, 0), is of the form R(t) for a unique rational number t.

2. Make a table listing the prime Gaussian integers π with norm N(π) ≤ 50.

3. The Eisenstein integers are the complex numbers of the form

\frac{x + y\sqrt{−3}}{2}

where x ≡ y (mod 2).

(a) Show that the sum and the product of two Eisenstein integers are also Eisenstein integers.

(b) If z is an Eisenstein integer and \overline{z} its complex conjugate, show that z + \overline{z} and z\overline{z} are both rational integers.

(c) Find all Eisenstein integers that divide 1 in the ring of Eisenstein integers.

(d) Show that 2, 5, and 11 are Eisenstein primes, while 3, 7, and 13 are not.

Document network location:

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