# Tagged Questions

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 ...

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### Under which conditions this condition of normality of groups translates into normality of extensions

Let $E/K$ be a Galois extension and $H \leq Gal(E/K)=: G$ be a subgroup. Under which hypothesis is the following true: $H$ is normal in G if and only if $E^{H}/K$ is normal. This looks similar to a ...
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### Dihedral groups are solvable [duplicate]

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ...
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### How to solve in radicals this family of equations for any degree $k$?

Part I. Given any constant $a,b$, the equation in $x$, $$\left(\frac{x+\sqrt{x^2+4a}}{2}\right)^{k}+\left(\frac{x-\sqrt{x^2+4a}}{2}\right)^{k}=b\tag1$$ is solvable in radicals for any degree $k$. ...
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### Let $F$ be the splitting field of $\psi_{12}(X)$ over $\mathbb{Q}$, with $\psi_{12}$ the cyclotomic polynomial.

So, we know that $\operatorname{Gal}(F/\mathbb{Q}) = \{\operatorname{id}, \sigma_1 , \sigma_2 , \sigma_3 \}$, with $\sigma_1 (\xi) = \xi^5$, $\sigma_2 (\xi) = \xi^7$, $\sigma_3 (\xi) = \xi^{11}$. I ...
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### Multiplication table of a Galois group?

I'm looking at the polynomial $x^4 − 4x^2 + 16$. I know that its roots are $1\pm\sqrt{3}$ and so its normal field extension is $\mathbb Q(i, \sqrt{3})$. However, I am also asked to give a ...
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### Faithfulness of Galois extensions of commutative rings

I need some help to understanding why Galois extensions of commutative rings are faithful. The definitions I'm using for Galois extensions is the one below. Let $R \rightarrow T$ be a ring ...
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### continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$

Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$ proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ ...
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### Quadratic extensions - understanding

If $[L:K]=2$ then $L/K$ is a quadratic extension. Can we think of a quadratic extension as $L=K(\alpha)$ where $\alpha$ is anything not in $K$(be it algebraic or transcendental)? This makes $L$ a ...
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### Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$

For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$? This problem appears to be quite common, however my understanding of Galois ...
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### Proof containing abelian groups.

Let $G \leq S_{999}$ be an abelian subgroup of order $|G| = 1111$. Prove that there exists $i \in$ {$1,2,...,999$} such that $\forall α \in G, α(i) = i$. Okay so I came across this problem and even ...
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### Show that the ring automorphisms of $\mathbb R$ are continuous

Prove that $-\frac{1}{m} < a -b < \frac{1}{m}$ implies $-\frac{1}{m} < \sigma a -\sigma b < \frac{1}{m}$ for every positive integer $m$. Conclude that $\sigma \in$ ...
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### $K$-homomorphisms from an étale $K$-algebra to a field

I am somewhat confused by the proof of Theorem 1.5.4 (p.22) (Grothendieck's version of the main theorem of Galois Theory) in Szamuely's Galois Groups and Fundamental Groups. The theorem establishes an ...
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### Galois Theoretic Proof of Fundamental Theorem of Algebra

The Galois Theory proof involves proving that $F(i)$ is algebraically closed for any real closed field $F$ (where $i$ is a square root of $-1$), where we may define a real closed field as an ordered ...
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### Explain how to compute $\cos(2\pi/13)$ by solving quadratic and cubic equations only

I know that we can express $2\pi/13$ as a root of unity on the unit circle taking $z^{13} = 1$ and $z=\cos(2\pi/13)+i\sin(2\pi/13)$ and that we should be able to find a polynomial for this over the ...
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### What are all possiblities for Galois group $G(L(\sqrt[5]{3})/L)$?

If $L$ is a subfield of $\mathbb C$ (which implies it has characteristic 0 and $\mathbb Q$ as its prime subfield) , what can Galois group $G(L(\sqrt[5]{3})/L)$ be? I start to solve this problem by ...
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### Galois Field to bits, implementation is fine, but the mathematics is not.

I am working on a hardware implementation of the SIMON cipher and the key expansion is based on GF(2). The original paper is here, https://eprint.iacr.org/2013/404 I have successfully created the ...
### Find the automorphisms for the Galois Group of the minimial polynomial $x^4+1$.
Determine the splitting field $L$ for this polynomial over $\mathbb{Q}$. The splitting field of $x^4+1$ must contain the solutions to $$x^4+1=0,$$ that is, $x^4=-1$. So $x^2=\pm i$, and ...
### Why $Gal(\mathbb Q(\zeta _n)/\mathbb Q)\hookrightarrow (\mathbb Z/n\mathbb Z)^\times$
Let $\zeta _n=e^{\frac{2i\pi}{n}}$ an let $\mu_n=\{1,\zeta _n,\zeta _n^2,...,\zeta _n^{n-1}\}$. 1) Show that $\mathbb Q(\mu_n)=\mathbb Q(\zeta _n)$ and that $\mathbb Q(\zeta _n)/\mathbb Q$ is ...