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