Degree of Field Extension

[Tsakanikas Nickos]

Let \( \displaystyle \left( p_{n} \right)_{ n \in \mathbb{N} } \) be the sequence of prime numbers. For each \( \displaystyle n \in \mathbb{N} \)

consider the field extension \( \displaystyle \mathbb{Q} \left( \sqrt{ p_{1} } , \ldots , \sqrt{ p_{n} } \right) \) of \( \displaystyle \mathbb{Q} . \)



(i) For each \( \displaystyle n \in \mathbb{N} \) compute the degree of the above field extension of \( \displaystyle \mathbb{Q} \);

That is, find

\[ \displaystyle \left[ \, \mathbb{Q} \left( \sqrt{ p_{1} } , \ldots , \sqrt{ p_{n} } \right) \, : \, \mathbb{Q} \, \right] , n \in \mathbb{N} \]



(ii) Show that

\[ \displaystyle \left[ \, \mathbb{Q} \left( \sqrt{ p_{1} } , \ldots , \sqrt{ p_{n} } , \ldots \right) \, : \, \mathbb{Q} \, \right] = \infty \]