How Do Units Determine Norm Values in Rings?

[Jupiter]

How can I show that if
$$\frac{a}{a^2-2b^2},\frac{b}{a^2-2b^2}\in \mathbb{Z}$$

then $$a^2-2b^2=\pm 1$$?
If you care to see the whole problem, you can find it here:
http://www.math.rochester.edu/courses/236H/home/hw12.pdf
It's #4 part c.

BTW, why is the significance of this "norm map"? I tried to google it for fun, but couldn't find much.

 

[Jupiter]

I got it! I needed part b. I was going about the problem the wrong way. Thanks anyway!

 

[M98Ranger]

To show that a^2-2b^2=\pm 1, we can use the fact that the elements a and b are units in the ring \mathbb{Z}, which means they have multiplicative inverses. This means that there exist elements c and d such that ac=ca=1 and bd=db=1. We can then rewrite the given expressions as:

\frac{a}{a^2-2b^2} = \frac{1}{a-2b^2a} = c
\frac{b}{a^2-2b^2} = \frac{1}{a-2b^2a} = d

Multiplying these expressions together, we get:

\frac{ab}{(a^2-2b^2)^2} = cd

Since cd=1, we can rewrite this as:

\frac{ab}{(a^2-2b^2)^2} = 1

This can be rearranged to give us:

a^2-2b^2 = \pm 1

This shows that if the given expressions are elements of \mathbb{Z}, then a^2-2b^2 must equal \pm 1.

Now, for the significance of the "norm map", it is a function that maps elements of a ring to the set of integers. In this case, the norm map is defined as:

N(a+bi) = a^2-2b^2

This map helps us understand the structure of the ring \mathbb{Z} (the Gaussian integers, which are numbers of the form a+bi, where a and b are integers). By using the norm map, we can show that the elements a and b are units in the ring if and only if N(a+bi)=\pm 1. This is exactly what we showed above, which is why the norm map is significant in this problem.