Is Q(√2,√3)=Q(√2+√3) ?

Q(√2,√3)={a+b√2+c√3+d√6∣a,b,c,d∈Q}

Q(√2+√3)={a+b(√2+√3)∣a,b∈Q}

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

How to conclude from there?

**Answer**

Q(√2+√3)⊆Q(√2,√3) is clear.

Now note that (√2+√3)−1=1√2+√3=√2−√32−3=√3−√2 hence √3−√2∈Q(√2+√3) and hence

√2+√3+√3−√2=2√3∈Q(√2+√3) and hence √3∈Q(√2+√3). Note that by a similar argument you get √2∈Q(√2+√3) and hence Q(√2,√3)⊆Q(√2+√3).

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