We have started with group theory this week and we have got a set of exercises... I have done the first 10 exercises, but after the 10th one they get so much harder and I have no clue how to solve this one. The next 10 exercises are based on this one...
I'd really appreciate if anyone could give me a hand. Thanks!
"Show that Z[(3)^(1/2)] has infintely many units. (Z is the integer set...)
Call a non-unit x in Z[(-1)^(1/2)] (Z[(-2)^(1/2)], Z[(-3)^(1/2)], Z[(3)^(1/2)] respectively) irreducible if
0 /= (not equal to) x /= y*z
for any two non-uunit y and z in Z[(-1)^(1/2)] (Z[(-2)^(1/2)], Z[(-3)^(1/2)], Z[(3)^(1/2)]
Note: I think Z[(-1)^(1/2)] would be the set of all complex numbers of the form a+b(-1)^(1/2). The elements of Z[(-1)^(1/2)] are called Gaussian integers... but I don't know if this has much to do with the actual question being asked :-/
For questions involving these types of rings, it is often helpful to define a norm map. Define by . You can check (and this is important) that defined this way, is a multiplicative map: that is, for any , . (Check this by straightforward computation.)
Then one can show that is a unit if and only if .
Once we have that, what we need is a unit whose absolute value is larger than 1. Take the element . We see that , so by the above statement, is a unit (indeed, ). Since (the standard absolute value function on ), we know from basic properties of real numbers that so, in particular, .
But since is a multiplicative map, . Again by the above statement, we conclude that must also be a unit in (you can actually compute the inverse, if you really want).
Iterating this process gives you an infinite list of units: .