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I have that $f(x)$ is irreducible over $\mathbb{Q}$ but i tried to apply the theorem 24.4.9 in Fundamentals of abstract algebra by malik

Theorem $24.4 .9$ Let $f(x)$ be an irreducible polynomial in $\mathbb{Q}[x] .$ Suppose that $deg f(x)=p$, where $p$ is a prime. If $f(x)$ has exactly $p-2$ real roots and two complex toots, then the Galois group of $f(x)$ over $\mathbb{Q}$ is $S_{p}$.

but $f'(x)$ has 4 real roots then $f(x)$ don't have complex roots, so I have no idea what to do because I think that the exercise may be wrong since it could be seen that the galois group is $S_5$ which is not soluble. Any idea?

Gustavo Andres Pava Parra
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    According to GAP, the Galois group is the solvable Frobenius group $C_5\rtimes C_4$. I don't have a solution by hand at the moment. – Brauer Suzuki Sep 04 '21 at 05:04
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    Hint: $f(x)=(x-1)^5-2$ so it shares the splitting field with $g(x)=x^5-2$. – Jyrki Lahtonen Sep 04 '21 at 05:08
  • Ok I'll try to solve it, I didn't think about rewriting $f(x)$ like that. Thanks. – Gustavo Andres Pava Parra Sep 04 '21 at 05:27
  • @JyrkiLahtonen Why are you answering in a comment? – Arthur Sep 04 '21 at 05:52
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    @Arthur We have covered the Galois group of $x^p-2$, $p$ a prime, already. And undoubtedly also $x^5-2$ in particular. I won't post answers to duplicates. Even if I don't have the time to search for the best possible target. – Jyrki Lahtonen Sep 04 '21 at 05:55
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    @JyrkiLahtonen If you think about why you won't post answers to dupes, doesn't that doubly apply to answers in comments to dupes? In that it makes it even more difficult to find all the answers to a given frequent question? In my opinion, if you don't want dupes to have answers in answer posts, you really shouldn't want dupes to have answers in the comment section. – Arthur Sep 04 '21 at 05:58
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    @Arthur Possibly? It might be a good idea to discuss this in detail in meta. My thinking (or a gut feeling) is that when a question gives no chance to add new content to the site, it may still be ok to give the asker a hint. Putting my teacher's hat on in a sense. True, searching for a duplicate would still be better. Particularly when the hint pretty much spoils the question, which was the case here. It is possible that I also have selfish motives that I cannot identify myself. My subconscious is not very reliable. – Jyrki Lahtonen Sep 04 '21 at 06:04
  • Related 1, 2, 3, 4. – Jyrki Lahtonen Sep 04 '21 at 10:59
  • @Arthur, I probably (?) feel strongly that the most offensive aspect of rampant dupe answering is doing it to rake in rep. Effectively trying to get paid twice, or ten or a hundred times for the same work. Answering in a comment is a form of rep denial. Also, now I found a list of duplicate candidates. Let's see what the others think. – Jyrki Lahtonen Sep 04 '21 at 11:07
  • @JyrkiLahtonen For me answering questions that turn out to be dupes (or answering questions here at all, really) was never actually about the rep. Always just another opportunity to put my teacher's hat on. I always thought answering in the comments for worse damage to the site than answering dupes. Because they don't show up as answers, they aren't searchable, and they sidestep the peer vetting system we have here. – Arthur Sep 04 '21 at 11:20

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