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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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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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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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
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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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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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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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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
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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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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
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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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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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2answers
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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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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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2answers
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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
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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
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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
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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
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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 ...
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2answers
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Whether or not an extension is Galois over $\mathbb Q$

Is the extension $\mathbb Q(\sin\frac{2\pi}n)$ a Galois extension over $\mathbb Q$? For the case when $n$ is divisible by $4$, I know that this happens. But I don't know how to do this in general. ...
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Find a second root of $x^3+px+q$ given the first root

This is a problem from Artin where given one root $a$, you have to find an equation for a second root in terms of $a$, $p$, $q$, and the square root of the discriminant $\delta$. Here's what I have ...
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Compute the splitting field and the Galois group of $x^4 - 5$ over $\mathbb{Q} (\sqrt{5})$. [duplicate]

I believe the splitting field is easily found by the following. $x^4 - 5 = (x^2 - \sqrt{5})(x^2 + \sqrt{5})$, so the splitting field is $\mathbb{Q}(\sqrt[4]{5},i)$, or is this incorrect? Once I ...
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1answer
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Weakly normal polynomials and normal polynomials.

I have been going through the notes of Prof. Pete L. Clark here (warning: long pdf). They are rough notes on Field theory and on page 30 he defines $P \in K[t]$ a normal polynomial if $P \in L[t]$ is ...
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Tower within a Galois extension

Consider the following tower of fields: $$ K \subset M \subset L $$ If $ L/K$ is a finite Galois extension, then is it true that $ M/K $ is a Galois extension ? Is it also finite ? It is clear to ...
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Let $F/k$ be a Galois extension. Show there exists an $\alpha \in F$ such that $\{ \sigma(\alpha) | \sigma \in G (F/k)\}$ is a basis for $F$ over $k$.

Title. I need to show that $\{ \sigma(\alpha) \mid \sigma \in G (F/k)\}$ forms a vector space for $F$ over $k$ if $F/k$ is a Galois extension. I know that if $F/k$ is Galois, then $F$ is the ...
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Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
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1answer
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Automorphism of $\mathbb{Q}({\zeta_n})/\mathbb{Q}$

I came across the theorem where, for $n=p^{a_1}\cdots p^{a_m}$: $Gal(\mathbb{Q}({\zeta_n})/\mathbb{Q})\simeq$ $Gal(\mathbb{Q}({\zeta_{p^{a_1}}})/\mathbb{Q})\times ...
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What are all the intermediate fields of $\mathbb{Q}\big(\sqrt{3+\sqrt{5}}\big)$ containing $\mathbb{Q}$?

I've come to a fork in the road, and it is sending me on wild goose chases. This question comes from a final exam for an Intermediate Abstract Algebra course I just took this past Spring. I'm ...
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1answer
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Frobenius automorphism and cyclotomic extension

There is a lemma from a lecture I attended where I scribbled down notes and try to make sense of the proof afterwards and there is a spot at which I am stuck. First I have to set up some notations and ...
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1answer
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Isomorphisms in characterisation of Galois extension

My definition of an extension $M/K$ to be Galois is that $Gal(M/K)$ only fixes things in K. I'm trying to prove that this is equivalent to $M/K$ being normal and separable. I know that fact that if ...
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1answer
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Why is $Gal(\Omega/k)$ a topological group under the Krull topology?

For an infinite Galois extension $\Omega/k$ the Krull topology on $G:=Gal(\Omega/k)$ is defined by taking as a basis for the neighbourhood of an element $\sigma \in G$ all cosets of the form $\sigma ...
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Constructing a tower to the splitting field of an arbitrary irreducible cubic.

Let f(x) = x3 + ax2 + bx + c ∈ $\mathbb{Q}$[x]. Let K be the splitting field of f(x). I want to construct a tower: $\mathbb{Q}$ ⊂ K1 ⊂ ... ⊂ Kr = K Each Ki = Ki-1(α) where either ...