Exercise 8.3.5 (Fundamental theorem isomorphism)

Question

Choose one of $\mathbb{Q}(\xi^2), \mathbb{Q}(i) or \mathbb{Q}(\xi^2i)$ and do the same as I did for it as I just did for $\mathbb{Q}(\xi^2,i)$.

Hassaan Naeem · Accepted Answer

\textbf{Solution:}
We choose $L=\mathbb{Q}(\xi^2)$. 
This gives us $$ G / \langle k, \rho^2 \rangle  \cong \text{Gal}(\mathbb{Q}(\xi^2):\mathbb{Q}) $$
The left hand side is the quotient of $D_4$ by a subgroup isomorphic to $C_2 \times C_2$. As we can observe, it has order 2.
Hence $G/ \langle k, p^2 \rangle \cong C_2$. On the other hand, $\mathbb{Q}(\xi^2)$ is the splitting field over $\mathbb{Q}$ of
$t^2 -2$. This is due to the fact the $\xi^2 = (\sqrt[4]{2})^2 = \sqrt2$. We know trivially that $\text{Gal}_{\mathbb{Q}}(t^2-2) = \text{Gal}(\mathbb{R}: \mathbb{Q}) \cong C_2$.
This confirms the isomorphism.

Exercise 8.3.5 (Fundamental theorem isomorphism)

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Exercise 8.3.5 (Fundamental theorem isomorphism)

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