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Let $ K = \mathbb{Q}(\sqrt[5]{2}, \zeta_5) $, where $ \sqrt[5]{2} $ is a real fifth root of 2 and $ \zeta_5 $ is a primitive fifth root of unity. The field $ K $ is the splitting field of $ f(x) = x^5 - 2 $ over $ \mathbb{Q} $, and its Galois group $ \text{Gal}(K / \mathbb{Q}) $ has order 20.

I want to determine all normal intermediate fields $ F $ of $ K $ over $ \mathbb{Q} $.

Here are the normal intermediate fields I have identified so far:

  1. $ K $ itself, the splitting field of $ x^5 - 2 $.
  2. $ \mathbb{Q}(\zeta_5) $, the field generated by a primitive fifth root of unity

Are there any other normal intermediate fields of $ K $ over $ \mathbb{Q} $? If so, how can I systematically find them and describe them explicitly?

Any assistance or insight would be greatly appreciated.

Hafeez1729
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    The systematic method is to compute the Galois group, and determine its normal subgroups. You're missing $\mathbb Q(\sqrt 5)$, which is a subfield of $\mathbb Q(\zeta_5)$. – Mathmo123 Jan 16 '25 at 03:17
  • See Tom Oldfield's answer to a more general question or here for the same. May be also this? – Jyrki Lahtonen Jan 23 '25 at 05:34
  • When you understand the Galois group, it becomes a matter of identifying the normal subgroups. How comfortable you are with working on small groups like this? I'm asking because right around this point in your studies lack of group theory begins to get in the way of working out examples in Galois theory. I face that exact problem when trying to guide the students into Galois theory, and we are pressed for time! – Jyrki Lahtonen Jan 23 '25 at 05:39
  • For example with a suitable background you can quickly tell that this group of order $20$ has a unique (hence normal) subgroup of order five. Its field of fixed elements is on your list already. Correspondence theorem then tells right away, that there is a unique subgroup of order ten. Similarly, subgroups of order four (five of them) are each other conjugates. Hmm.... another approach may be cleaner :-) – Jyrki Lahtonen Jan 23 '25 at 05:44

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