Topology of Numbers · Problem 8.2.4

Quadratic Fields — Allen Hatcher

Mathematics — Number Theory · unique-factorization-euclidean-algorithm

Problem

In this problem we consider

\mathbb{Z}[\sqrt{-2}]

, so elements of

\mathbb{Z}[\sqrt{-2}]

are sums

x + y\sqrt{-2}

for integers

x

and

y

, with

N(x + y\sqrt{-2}) = (x + y\sqrt{-2})(x - y\sqrt{-2}) = x^2 + 2y^2

. (a) Draw the topograph of

x^2 + 2y^2

including all values less than

70

(by symmetry, it suffices to draw just the upper half of the topograph). Circle the values that are prime (prime in

\mathbb{Z}

, that is). Also label each region with its

x/y

fraction. (b) Which primes in

\mathbb{Z}

factor in

\mathbb{Z}[\sqrt{-2}]

? (c) Using the information in part (a), list all primes in

\mathbb{Z}[\sqrt{-2}]

of norm less than

70

. (d) Draw a diagram in the

xy

-plane showing all elements

x + y\sqrt{-2}

\mathbb{Z}[\sqrt{-2}]

of norm less than

70

as small dots, with larger dots or squares for the elements that are prime in

\mathbb{Z}[\sqrt{-2}]

. (There is symmetry, so the primes in the first quadrant determine the primes in the other quadrants.) (e) Show that the only primes

x + y\sqrt{-2}

\mathbb{Z}[\sqrt{-2}]

with

x

even are

\pm\sqrt{-2}

. (Your diagram in part (d) should give some evidence that this is true.) (f) Factor

4 + \sqrt{-2}

into primes in

\mathbb{Z}[\sqrt{-2}]

. (g) Use the unique factorization property in

\mathbb{Z}[\sqrt{-2}]

to determine which numbers are represented by the form

x^2 + 2y^2

, as was done in the text for

x^2 + y^2