# Questions tagged [galois-theory]

Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-connections).

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### How to show that the field $\mathbb{F}_2(t^{\frac{1}{3}})$ does not contain a third root of unity

I want to find a non-normal extension in characteristic p. The following extension is not normal : $$\mathbb{F}_2(t)\subset \mathbb{F}_2(t^{\frac{1}{3}})$$ It's minimal polynomial is : $X^3-t$ and it'...
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### How can we subtract elements of the unit group so that the difference is still in the unit group

I am trying to find the rank of a matrix formed by the characters of a finite ring over the rationals. The order of the matrix is the number of elements in the unit group by the number of elements in ...
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### Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
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### Yes/No .An extension having Galois group of order $1$ is normal [closed]

Is the following statement true/false ? An extension having Galois group of order $1$ is normal. I think this statement is false take $K= \mathbb{Q}(\sqrt[5]{3}):\mathbb{Q}$ Galois$(K)=\{e\}$ ...
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### Confusion on isomorphism profinite completion integers

Question: Prove that $\hat{\mathbb{Z}} \cong \prod_{p} \mathbb{Z}_{p}$ is an isomorphism of topological rings. Own attempts: Although this specific question has been asked quite a few times here, ...
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### Searching Explanation/References for a certain Corollary to Chebotarev Density Theorem

I am reading “Notes on the model theory of finite and pseudo-finite fields” by Zoé Chatzidakis https://www.math.ens.psl.eu/~zchatzid/papiers/Helsinki.pdf and on page 16 (or in this newer version of ...
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### The Frobenius Endomorphism is Surjective iff the field is perfect

I'm taking a Galois theory class right now. I've read and understood the proof that the Frobenius endomorphism is surjective iff the field is perfect (working in characteristic $p$). But it just feels ...
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### Why does $x^3-7$ have Galois group isomorphic to $S_3$? [duplicate]

I'm not concerned with showing that the order of the Galois group is $6$; I've already done that. I'm more concerned with the structure of the Galois group. So $x^3-7$ has the roots ...
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### Cohomology class of automorphism group of Galois form

Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
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### "Galois theory" on graphs

Let $G = (V, E)$ be a graph. For the sake of simplicity, let us assume $G$ is undirected and finite. The automorphism group $\mathrm{Aut}(G)$ contains all graph isomorphisms $\phi : G \rightarrow G$. ...
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### Subfields of splitting field of $x^4+25$ over $ℚ$.

Let $F$ be the splitting field of the polynomial $x^4+25$ over $ℚ$. List all subfields in $F$ and the corresponding subgroups in the Galois group. is problem $1$ on this pdf. The solution is: As we ...
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### Assistance with an exercise on field endomorphisms

Working through the problems in a book on field theory (Field Extensions and Galois Theory by Bastida). I came across one which I thought looked like a "routine" exercise, but has been ...
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### Automorphism group of $x^3 - x - y^2$.
Let $k$ be some field of characteristic $\neq 2$. The polynomial $f(x,y) = x^3 - x - y^2$ is irreducible in $k[x,y] \cong k[x][y]$ as it cannot have a root in $k[x]$. By Gauss's lemma, $f$ is then ...