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I do not understand very well about extensions of fields $\mathbb{Z}_p$. I understood about field's extensions of $\mathbb{Q}$, for instance, $\mathbb{R},\mathbb{C}$ etc.

But, what fields extend $\mathbb{Z}_p$?

Because of these uncertaintes, I cannot solve the following problem:

Determine the Galois group of the extension $\Sigma:\mathbb{Z}_2$, where $\Sigma$ is the splitting field of $f(x)=x^4+x+1\in\mathbb{Z}_2[x]$ over $\mathbb{Z}_2$.

amWhy
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Quiet_waters
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    I presume $\Bbb Z_2$ denotes the field of order $2$, not the ring of $2$-adic integers? – Angina Seng Jul 17 '18 at 17:45
  • Yes, maybe I should use $\mathbb{Z}/2\mathbb{Z}$, right? Thanks. – Quiet_waters Jul 17 '18 at 17:45
  • I would use $\Bbb F_2$. – Angina Seng Jul 17 '18 at 17:46
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    $\Bbb{Z}_2$ is IMHO fine, given the context. Some prefer $\Bbb{F}_2$, some use $GF(2)$. Anyway, all extensions of finite fields are Galois. Have you shown that $f(x)$ is irreducible over $\Bbb{Z}_2$? – Jyrki Lahtonen Jul 17 '18 at 17:47
  • "But, what fields extend $\mathbb{Z}_p$?" Probably none that you heard of before you started field theory. This will likely be the first time you encounter them, and you will just have to use their definition (as some specific extension of $\Bbb Z_p$) to do calculations in them. For instance, the second degree extension of $\Bbb Z_2$ has four elements $0, 1, a, a+1$, with multiplication is given by $a^2 = a+1$. – Arthur Jul 17 '18 at 17:51
  • Actually, you can take a look at the splitting field $\Bbb{F}_{16}$ of $f(x)$ over $\Bbb{F}_2$ here. I prepared that example for referrals. It does not do exactly what you need here, but you do get a feel for the arithmetic :-) Anyway, we have covered the Galois theory of extensions of finite fields many times over (which is why I'm less than thrilled to spell out all the details for you). Have you searched within [tag:finite-fields]? – Jyrki Lahtonen Jul 17 '18 at 17:51
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    Check out Lemma 2.1 of the following: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/finitefields.pdf – Morristewgray Jul 17 '18 at 18:20
  • Do you understand the field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3})$? – Eric Towers Jul 17 '18 at 19:00
  • About notation, I didn't know the notation $\mathbb{F}_p$, I thank you for the help. – Quiet_waters Jul 17 '18 at 20:50
  • @EricTowers, yes, this type of extension (of $\mathbb{Q}$ and similar) I've studied before. – Quiet_waters Jul 17 '18 at 20:53
  • @Arthur, now I'm beginning to understand. Thanks. – Quiet_waters Jul 17 '18 at 20:54
  • @JyrkiLahtonen, yes, I've proved the irreducibility before. About the second comment, I thanks very much, I'm studying. – Quiet_waters Jul 17 '18 at 21:00
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    @Morristewgray, thanks very much, the document cited has helping me with more than one exercise that I'm solving. – Quiet_waters Jul 17 '18 at 21:07

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Check out the following document: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/finitefields.pdf

Morristewgray
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    Could you please expand this a bit vian an [edit]. Answer posts should be actual answers not links to resources that provide the answer. – quid Jul 17 '18 at 18:24
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    Links go stale. Links do not contain keywords relevant to searches by future learners. – Eric Towers Jul 17 '18 at 18:25