I have some issues with this exercise. How should I use induction on k? Maybe someone can give me a hint :)
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The hint is pretty clear. Have you tried to perform the induction step ? Have you tried for small values of $k$, such as $k=1$ and $2$ ? – GreginGre Sep 23 '20 at 11:19
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Hi again! I have tried the induction step with k=1, i.e I want to show that $[F(\sqrt{d_1}:F]=2$. The minimal polynomial for for $\sqrt{d_1}$ over F is $x^2-d_1$. $\sqrt{d_1}$ is a root in this polynomial, and is of degree 2, so we get that $[F(\sqrt{d_1}:F]=2$, but I am unsure if that's the correct way to do it, because in the hint my teacher says that it is convenient to see K as an extension of $F(\sqrt{d_1}$, but I haven't used that, and I don't know for what I should use that. But then I move on and assume that it is true for k, i.e $[F(\sqrt{d_1},...,\sqrt{d_k}):F]=2^k$ – Sumi Sep 23 '20 at 16:25
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And then i gotta show it for k+1, i.e I want to show that $[F(\sqrt{d_1},...,\sqrt{d_{k+1}}):F]=2^{k+1}$, but for this I am a bit unsure what to do. – Sumi Sep 23 '20 at 16:29
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See this answer in the dupe. – Bill Dubuque Sep 23 '20 at 18:27
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Hi! The answer in the link is not understandable for me and I feel like it is confusing to question it in that thread. – Sumi Sep 24 '20 at 07:39