Proving Equality of Fields with Distinct Primes in Z

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Hey,
Does anyone know how to show that these fields are equal:
Q(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_k})=Q(\sqrt{p_1}+\sqrt{p_2}+...+\sqrt{p_k}),
where p_1,...,p_k are distinct primes in Z.

One inclusion is clear to me, but I'm having problems showing they're equal. Thanks!
 
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Induction on k, maybe?
 
Right...note to self, always remember about induction. Thank you.
(although my solution using induction looks super messy)
 

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