Proving Equality of Fields with Distinct Primes in Z

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The discussion centers on proving the equality of fields defined as Q(√p₁, √p₂, ..., √pₖ) and Q(√p₁ + √p₂ + ... + √pₖ), where p₁, p₂, ..., pₖ are distinct primes in Z. One participant suggests using induction on k to demonstrate this equality, acknowledging that their initial approach appears convoluted. The conversation highlights the importance of induction in field theory and the challenges associated with proving such equalities.

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T-O7
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Hey,
Does anyone know how to show that these fields are equal:
[tex]Q(\sqrt{p_1},\sqrt{p_2},...,\sqrt{p_k})=Q(\sqrt{p_1}+\sqrt{p_2}+...+\sqrt{p_k})[/tex],
where [tex]p_1,...,p_k[/tex] 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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