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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Prove the form of a fixed field

let $C_3=\langle\sigma\rangle$ and let $\sigma$ act on $K(s,t)$ by $s \mapsto t$ and $t \mapsto -s-t$ I want to prove that $K(s,t)^{C_3}=K(u,v)$ with $u=\frac{s^2+t^2+st}{st(s+t)}$ and ...
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Compositum of normal extensions is a normal extension

I'm trying to prove that if $ F \subset K, F \subset M $ are normal extensions, $ K,M \subset E $, then $ KM$ is also a normal extension of $ F $. I tried using the fact that $ F \subset KM $ is a ...
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1answer
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Determine the Galois group of $ F(x^5) \subset F(x) $

I'm rather new to Galois theory and have been given this exercise: Suppose $ F $ is respectively equal to $ \mathbb{Q}, \mathbb{C}, \mathbb{F}_5 $ (the third one is just the 5-element field). My task ...
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Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
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Summary of Galois quintic unsolvability

Is this a sufficient flow of logic? Summary: Consider a fifth degree polynomial that has a formulaic solution. Then we have a radical extension with a string of subgroups of the Galois group which ...
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1answer
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Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is an algebraic and infinite extension of Q

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is a algebraic and infinite extension on Q. Well, if i consider for every p prime, the polynomial p(x)=x^2−p, then p(x) is in Q(p√∣p is ...
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Galois of $x^3+2$

the splitting field of $x^3+2$ is $E=\mathbb Q(\sqrt[3]{2},i\sqrt{3}$) and I know the $Gal(E:\mathbb Q)$ has six elements which is isomorphic to $S_3$. How to find all intermediate subfields using ...
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Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
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Galois theory question (Kummer's Theory)

This question is from a past exam paper. Suppose $E = \mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3} - 39X + 26 \in \mathbb{Q}[X]$. Prove that $E$ is cyclic Galois extension of ...
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Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
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1answer
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If a field extension contains a cyclotomic extension is it solvable?

This doesn't seem like it should be true but I'm not entirely sure - I would appreciate anyone looking over the following: If we have a finite Galois field extension $L/K$ and $M \subset L$ a ...
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1answer
40 views

Prove the Galois Group is Isomorphic to $S_3$

Prove G=$Gal(\mathbb{Q}(\sqrt[3]{2},i\sqrt{3}) : \mathbb{Q})$ is isomorphic to $S_3$ I know that the G has 6 automorphims, and $S_3$ has order 3! then consider polynomial $x^3-2 = ...
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$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
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1answer
47 views

Transitivity of the discriminant of number fields

Let $M/L/K$ be a tower of number fields with discriminant of $M/K: d_M$ and of $L/K: d_L$. I would like to find a transitivity theorem for the discriminant and by letting $p_i$ and $q_i$ be integral ...
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Ramification in $\mathbb Q(\zeta_5, \sqrt[5]2)/\mathbb Q(\zeta_5)$

Let $F=\mathbb Q(\zeta_5,\sqrt[5]2)$ and $K=\mathbb Q$ where $\zeta_5$ is a primitive $5$th root of unity and let $p=73$ be a prime in $K$. Fix primes $\mathfrak p$ and $\mathfrak q$ above $73$ in ...
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1answer
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What is the automorphism group of the field $\mathbb{Z} /p\mathbb{Z}(t)$?

Here, $t$ is transcendental over $\mathbb{Z} /p\mathbb{Z}$. How big is this group? What are its elements? Is for example the map $t \to -t$ an automorphism?
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1answer
50 views

Automorphisms (in the context of Galois Theory)

Can someone give an explanation of what an automorphism is, in the context of Galois Theory? I keep thinking it is the set of maps which send roots of polynomials to their conjugates but I feel that ...
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1answer
56 views

Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
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1answer
33 views

Galois Group, Field Extension Prove Abelian

Let F be an extension of $\mathbb{Q}$ and let $\omega = \cos{(\frac{2\pi}{n})}\sin{(\frac{2\pi}{n})}$. Prove that $Gal(F(\omega):F)$ is abelian. I am looking for a sketch of this proof. so far in ...
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Symmetric polynomials and g non symmetric

If $g=x_1+2x_2+3x_3, s_1=x_1+x_2+x_2, s_2=x_1x_2+x_1x_3+x_2x_3$ and $s_3=x_1x_2x_3$ , write $x_1, x_2$ and $x_3$ in function of $g, s_1, s_2$ and $s_3$.
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Dimension of compositum of two fields, one of them Galois.

Let $L, K, F = L \cap K$ be fields such that $[L:F] , [K:F] < \infty$. $LK$ is defined as the compositum of the two fields(shortest field containing $K$ and $L$ in some algebraic closure). Also, ...
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Cohomologies of Galois group of field extension

Let $k\subset K$ be a finite Galois extension with Galois group $G=\text{Aut}_k\,K$. How to prove that $H_i(G,K)=H^i(G,K)=0$ for all $i>0$?
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1answer
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Degree of $\mathbb{Q}(\omega)/\mathbb{Q}$ where $\omega^{3}=1$

I am working through Rotman's Galois Theory, and I came across an example that confused me a bit. Here is a screenshot of the problem: I am not sure, why the degree of ...
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1answer
33 views

Primitive element theorem, simple extension

Let $X$, $Y$ be indeterminates over $F_2$, the finite field with 2 elements. Let $L = F_2(X, Y )$ and $K = F_2(u, v)$, where $u = X + X^2$, $v = Y + Y^2$. Explain why $L$ is a simple ...
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$A_n$ as a Galois group

I am looking for a clear, detailed proof that the alternating group $A_n$ is realizable as a Galois group over the rationals $Q$. (I have seen proofs using Hilbert's irreducibility theorem, but they ...
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1answer
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Is the compositum of $L_1$ and $L_2$ equal to $L_1[L_2]$?

In a course about Galois theory, there is the following definition : Let $L_1$ and $L_2$ be two subfields of the field $L$. We define the compositum $L_1L_2$ of $L_1$ and $L_2$ as the smallest ...
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Galois extension over power series fields

Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if ...
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1answer
26 views

Showing that $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ irreducible over $\mathbb{Q}(\sqrt[4]{5})$

I am unsure how to show that the polynomial $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ is irreducible over $\mathbb{Q}(\sqrt[4]{5})$. If it were reducible, it would have a root in ...
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61 views

Does an inseparable extension have a purely inseparable element?

Assume $K/F$ is an inseparable extension. Is it necessary that K contains an element $u \notin F$ that is purely inseparable over $F$? I also posted in MO.
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Whether an extension is separable or not

It is a well known conclusion that if $K/F$ is a finite separable extension, then $K^{Emb(K/F)}=F$ (By which I mean the subfield of K fixed by $Emb(K/F)$). I am wondering, whether the inverse is true. ...
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1answer
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How is this $K$-automorphism well-defined?

I'm currently reading Hungerford algebra's chapter about Galois theory, I cannot understand the following example Let $F=K(x)$ with $K$ any field. For each $a\in K$ with $a\neq 0$ the map ...
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The splitting field of $x^7 - 2$ and its Galois group

This question is more or less a "is it me or is it something wrong in the answer sheet"-question. In one of the previous exam sheets at my university, I am supposed to find the order of the Galois ...
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1answer
75 views

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$ where $p$ is a prime number. My thoughts are: I am lost My intuition says it has to be $ \frac{p-1}{2}$ and ...
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Galois Group of an Extension

Question: Determine the isomorphism type of $ \mathrm{Gal}\,(\mathbb{Q}(\sqrt[8]{2},i)/\mathbb{Q}(i)) $. $\\$ This amounts to finding isomorphisms that send $\mathbb{Q}(\sqrt[8]{2},i)$ to ...
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Extension Field of $\mathbb{Q}$ and its Galois group

How many elements are in $Gal(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}:\mathbb{Q})$?
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Subgroups of Galois group and intermediate fields lattice for $(x^3-2)(x^2-3)$

I am trying to systematically determine all subgroups of Galois group and intermediate fields for $(x^3-2)(x^2-3)$(over $\mathbb Q$). It's not hard to determine the Galois group of $(x^3-2)$ and ...
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1answer
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Galois extension of $\mathbb Q$ of degree $n$

I have a basic question in Galois theory. For any given natural number $n$ is there a Galois extension of $\mathbb Q$ of degree $n$? I want to show that there are splitting fields of polynomials in ...
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1answer
39 views

About the Galois group of $\mathbb{Q}[\sqrt{5},\sqrt{2+\operatorname{i}}]/\mathbb{Q}[\sqrt{5}]$

Some notations: let $K=\mathbb{Q}[\sqrt{5}]$, $N=\mathbb{Q}[\sqrt{5},\sqrt{2+\operatorname{i}}]$, $a=\sqrt{2+i}$, $b=-a$, $c=\sqrt{2-i}$, $d=-c$. I know that $N/K$ is normal because it decompose the ...
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1answer
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separable polynomial $ \bmod p$ (definition)

Given a polynomial $ f(x) \in K[x]$, where $K$ is a number field, we say that $f$ is separable if all its roots are distinct in an algebraic closure of $K$. Question: What does it mean a ...
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Transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements.

I need to show that transcendence base of $\mathbb{C}$ over $\mathbb{Q}$ has infinitely many elements. Since I do not know much about ordinals and cardinals, a proof based on algebra (rather than ...
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Solving polynomial equations using more than radicals

It is well known that one cannot solve every polynomial equation over $\Bbb Q$ using just radicals. In other words, let $A_n = \{x^n - a\mid a \in \Bbb Q\}$, $A = \cup_n A_n$ and $\bar{\Bbb Q}$ the ...
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Showing $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$

I want to show that $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$, but I'm not that familiar with computing Galois groups so I don't really know how to do this exercise. How do I approach ...
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(Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$

(Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$. This is my rough proof to this question. I was wondering if anybody ...
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2answers
31 views

Whether an embedding is an automorphism

Let $K/F$ be a field extension and let $\sigma$ be an embedding from $K$ into $K$ over $F$. If $K/F$ is algebraic, prove that $\sigma \in Aut(K)$. I know how to prove the case when $K/F$ is finite. ...
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Galois group of the simple extension

Let $K=Q(\sqrt3,\sqrt5)$. Show that the extension is $K/Q$ is simple and also Galois extension. Determine its Galois group. I showed that extension is simple because $K=Q(\sqrt{3}+\sqrt{5})$ But I ...
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1answer
51 views

Field extension of fixed field has degree greater than the size of the group

Let $K$ be a field, $G\leqslant\mathrm{Aut}(K)$ a (finite) group, and $K^G$ the fixed subfield of $K$. How exactly would you go about proving the following?$$[K:K^G]\geqslant|G|$$ For some reason I ...
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1answer
42 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
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1answer
52 views

Galois Theory and Splitting Fields

So I have an exam tomorrow and I think I'm rather prepared as far as the theory goes (I have the theorems in the book memorized, etc), but I am rather worried about any "concrete" questions I may get ...
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Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof ...
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1answer
24 views

Whether a field extension contains $i$.

For which values of n does the cyclotomic extension over $Q$ contain $i$? My guess is that this is precisely when n is divisible by 4. If n is divisible by 4,then i can show this quite easily. But is ...