I believe that the first equation should be ae+bcd=1. This is just a system of equations in the unknowns e and c which you can solve. Just be careful of the special cases a=0 and possibly b=0.
Can you help me prove that this ring is a field?
R = Q[ ] = { a+b ; a,b are rational}
So I tried:
We must show that every element of R is a unit. So for some element (a+b* ) there must be some (e+c* ) such that
(a+b* )*(e+c* ) = 1
so (ae+bed) + (be + ac)* = 1
so ae-bed = 1 and be+ac =0
but then I get stuck after that
If is a perfect square then and so is a field.
If is not a perfect square, then, is irrational.
In this case choose . Then
Now, using that is irrational, prove that
which implies that the system has a solution ( besides, unique ).
Fernando Revilla