Integral Domains and Fields

[hsong9]

Homework Statement

a) show that Q(√5i) = {r +s√5i | r,s in Q} is a subfield of C.
b)show that Z(√5i) = {n + m√5i | n,m in Z} is a subring of C and find the units.

The Attempt at a Solution

a)
Let a = r + s√5i, b = r - s√5i for a,b in Q(√5i).
a + b = 2r, ab = r^2 + 5s^2, and -a= -r - s√5i are all in Q(√5i),
so Q(√5i) is a subring of C.
Suppose that a != 0 in Q(√5i). If a = r + s√5i, this means r^2 - 2s^2 != 0 in Q
because √5i not in Q. But then 1/(r^2 - 2s^2) in Q, so the fact that ab = r^2 - 2s^2
implies that a^-1 = b * 1/(r^2 - 2s^2) exists in Q(√5i). Hence Q(√5i) is a subfiled of C.
Correct?

If correct, similarly, I could show Z(√5i) is a subring of C.
and the units are a^-1 = 1/a
I am not sure whether the units are right or not.

 

[mighty2000]

Yes, your solution for part a) is correct. For part b), you are also correct in showing that Z(√5i) is a subring of C. However, the units for Z(√5i) would be different from those in Q(√5i). In Z(√5i), the units would be any element of the form a + bi, where a and b are integers and a^2 + 5b^2 = 1. So the units would be 1, -1, 1 + √5i, -1 + √5i, 1 - √5i, and -1 - √5i.