Is it possible to have a degree-$1234567890$ extension of $\Bbb Q(\sqrt5)$?
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Is it possible to have a degree-$1234567891$ extension of $\Bbb Q(\sqrt5)$?
Is it possible to have a degree-$1234567890$ extension of $\Bbb Q(\sqrt5)$?
&
Is it possible to have a degree-$1234567891$ extension of $\Bbb Q(\sqrt5)$?
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Adjoining a root of $x^{2n}-5$ yields a degree $n$ extension over $\mathbb Q(\sqrt 5)$.