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Hi I really need help in how to determine sub fields $L$ of $\mathbb{C}$ over $\mathbb{Q}$ and finding $[L:\mathbb{Q}]$. The question is:

Find subfields $L$ of $\mathbb{C}$ that are splitting fields over $\mathbb{Q}$, find $[L:\mathbb{Q}]$

$$\begin{array}{ll} \mathrm{(i)} & t^8 + 1 \\ \mathrm{(ii)} & t^4 -4t^2 +1\\ \mathrm{(iii)} & t^3 -5\\ \mathrm{(iv)} & t^{17} - 1\\ \end{array}$$

Any help would be great!

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1 Answer 1

For example take $t^3 - 5$ the roots of this polynomial are: $\sqrt{5}, \xi \sqrt{5}, \xi^2 \sqrt{5}$. Here $\xi$ is the primitive root of unity of order $3$. Now you note that you need to add both $\sqrt{5}$ and $\xi$ to $\mathbb{Q}$, for the polynomial to split. Hence $L = \mathbb{Q}(\xi,\sqrt{5})$. Now if you take $F = \mathbb{Q}(\xi)$, then $[L:\mathbb{Q}] = [L:F][F:\mathbb{Q}]$. The extension $L/F$ is Kummer and $F/\mathbb{Q}$ is cyclotomic, hence $[L:F] = 2$ and $[F:\mathbb{Q}] = 2$.

The rest are very similar and most rely on cyclotomic extensions. Remember that if you add a primitive root of unity of order $n$, you'll get a field extension of degree $\phi(n)$, where $\phi$ is the Euler totient function.

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