Exercise 5.3.7 (Subfield lemma)

Question

Find an example of Lemma 5.3.6 where $[LL':L]=2$, and another where $[LL':L]=1$.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
If we let $L = \mathbb{Q}(\sqrt2)$ and $L' = \mathbb{Q}(\sqrt3)$, we then get $LL' = \mathbb{Q}(\sqrt2,\sqrt3)$.
Then $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}(\sqrt3)] = 2$.\\
If we let $L = \mathbb{Q}(\sqrt4)$ and $L' = \mathbb{Q}(\sqrt3)$, we then get $LL' = \mathbb{Q}(\sqrt4,\sqrt3)$.
Then $[\mathbb{Q}(\sqrt4,\sqrt3):\mathbb{Q}(\sqrt3)] = 1$.
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Exercise 5.3.7 (Subfield lemma)

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Exercise 5.3.7 (Subfield lemma)

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