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$p$ be a prime number such that $p\equiv3\pmod 7$ then $p$ generates a prime ideal in $Z[\zeta_7]$ where $\zeta_7$ is the primitive 7th root unity.

In the Galois Theory course, I found this problem in one of my lectures. I don't know the Class field theory and Algebraic Number Theory as of now. Please let me know if this problem can be done by Galois Theory. Also, what is the structure of prime ideals in $Z[\zeta_7]$?

peter a g
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MathCosmo
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  • What makes you think it can be done with Galois Theory? For example, is it a problem from a book/course on Galois Theory? – Sebastian Monnet Oct 02 '20 at 13:36
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    I may be wrong but I would think that the question would not appear in a context where at least the basics of algebraic number theory had not been covered. You don't really need class field theory here, but the question does serve as an example introducing CFT concepts. Can you prove that the minimal polynomial of $\zeta_7$ over $\Bbb{Q}$ remains irreducible over $\Bbb{F}_p$ (after reduction mod $p$)? That is the key. What it boils down to is that the coset of $3$ is a generator of the multiplicative group $\Bbb{Z}_7^*$. – Jyrki Lahtonen Oct 02 '20 at 13:56
  • http://www-users.math.umn.edu/~garrett/m/algebra/notes/07.pdf I solved your claim and also found this article where the solution is better than mine. I have one question if 3 is a generator of that multiplicative group then how are claiming that it is a prime ideal in $Z[\zeta}$. – MathCosmo Oct 02 '20 at 15:40
  • The result you want follows from this: https://math.stackexchange.com/a/3839736/232 – Qiaochu Yuan Oct 02 '20 at 19:36
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    A side note: I have answered the question about irreducibility of cyclotomic polynomials over a finite field more times than i care to admit :-/ This is possibly the most careful scrutinized in that a fellow user spotted a mistake in the original version :-) – Jyrki Lahtonen Oct 04 '20 at 04:39

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