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Integral Domains and Fields

  1. Apr 14, 2009 #1
    1. The problem statement, all variables and given/known data
    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.

    3. 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.
     
  2. jcsd
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