You can probably try to show that the set is linearly independent over the field . The result will follow immediately.
It seems to be fairly straightforward if you know some field theory. (But I haven't actually tried it, so I could be wrong.)
Wouldn't it be possible to somehow get rid of the roots by taking the inverses of the exponents while keeping the sum equivalent ? Dealing with powers seems simpler somehow.
This is just an idea, I guess it would be possible by messing around with logs and using their irrational properties.
I was dreading it
I meant, expressing your statement using a sum of powers instead of a sum of roots. Some changes would be required, but for instance :
- or even better -
Seems easier than dealing with somehow.
But this is not gonna be "elementary" anyway