Is Q(√2,√3)=Q(√2+√3)\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})?

Is Q(2,3)=Q(2+3) ?

Q(2,3)={a+b2+c3+d6a,b,c,dQ}

Q(2+3)={a+b(2+3)a,bQ}

So if an element is in Q(2,3), then it is in Q(2+3), because 6=23.

How to conclude from there?

Answer

Q(2+3)Q(2,3) is clear.

Now note that (2+3)1=12+3=2323=32 hence 32Q(2+3) and hence
2+3+32=23Q(2+3) and hence 3Q(2+3). Note that by a similar argument you get 2Q(2+3) and hence Q(2,3)Q(2+3).

Attribution
Source : Link , Question Author : Tashi , Answer Author : Rudy the Reindeer

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