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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Totien-Sum: why GCD( {n}/d, q/d) = 1; implies Sum{Totient(d/q) } = q

Have seen answer to this question. still don't understand.. Totient sum is defined: q = Sum(Totient (d) ); sum on all d : d|q More specific; The proof has these steps: 1. If d is a divider ...
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1answer
38 views

Why is the absolute value of this Gauss sum obvious?

I came across the Gauss sum discussed in the following post in a problem from my Galois theory course: http://mathoverflow.net/a/71282. Why exactly is the square of its norm obvious?
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3answers
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Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $E \subset \mathbb{R}$ a field containing $\mathbb{Q}$ such that the extension $E \vert \mathbb{Q}$ is Galois, and let $F := E(\sqrt{-1})$. Show that the extension $F \vert \mathbb{Q}$ is ...
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1answer
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Showing automorphism group is trivial

Let $f(x) \in k[x]$ be a separable polynomial of degree $n\geq 3 $ with Galois group isomorphic to $S_n$, and let $\alpha \in \bar{k}$ be a root of $f(x)$. a) Show f is irreducible (I have already ...
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2answers
44 views

Proving that a field is not a splitting field of any polynomial

I am trying to prove that $Q(2^{1/3})$ is not a splitting field of any polynomial $p(x) \in Q[x]$. I know I need to show that there does not exist any polynomial in $Q[x]$ that has all of its roots ...
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Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
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TO Find galois group of cubic pooynomial [on hold]

$\text {prove that galois group of }x^3-4x+1\ \text{is}\ S_{3}$?? i have no idea how to approach this problem .please help
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Find isomorphism between Kummer's field and field of n-th roots

Let $\mathbb{k}$ be the field of all the $n$th roots of $1$ and $\mathbb{F=k}(\alpha_1, ..., \alpha_n)$ is a Kummer's field over $\mathbb{k}$, where $\alpha^n_i=a_i \in \mathbb{k^*}$, $i=1,\dots,m$. ...
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2answers
35 views

Galois group of a quartic which is also a quadratic in $x^2$

A few weeks ago a professor of mine mentioned that the Galois group of a certain type of quartic polynomial is easy to calculate, and at the time it seemed obvious to me so I didn't ask why. Now i'm ...
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1answer
66 views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] ...
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Reference needed with derivation of the equations for the cubic and quartic and proof of impossibility of quintic equation

My background is an undergrad year long course in algebra which got me through to some basic finite field and field extension stuff (most of fraliegh or Ian Stewart's book are familiar to me if that ...
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1answer
90 views

Splitting field in R[X]

Suppose $f$ is the irreducible polynomial $x^4+x+1$ in $\mathbb{Q}[X]$ and we will call $E_f$ the splitting field of $f$ (which is a subfield of $\mathbb{C}$). Suppose $\alpha$ is a complex root of ...
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1answer
65 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
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2answers
20 views

Irreducible polynomials over $GF(4)$.

I'm working with $GF(4)$ and I'm looking for irreducible polynomials of different degrees over that field. So $GF(4) = \{0,1,\alpha, \alpha + 1\}$ where $\alpha^{2} + \alpha + 1 = 0$, and first I'm ...
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1answer
25 views

how would you show that field automorphisms fix prime subfields?

Suppose K is a prime subfield of E, then if $\phi$ is an automorphism from E to E, we have for all x $\in$ K, $phi(x) = x$. I feel like this is just the definition of a field automorphism, but my ...
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1answer
34 views

Describing a Galois group for a field with the given roots adjoined

In this problem $\mathbb{Q}$ is viewed as a subfield of the complex numbers and a root $a = \sqrt[3]{3}$ is given. I find the min. polynomial for this root, which is $x^3 - 3$ and I factor it into: ...
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1answer
41 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
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1answer
14 views

Problem defining morphism in Galois cohomology of algebraic group

Let $K$ be a field and $G$ an algebraic group defined over $K$. If $M\supseteq L$ are two finite Galois extensions of $K$, then the groups $\text{Gal}(M/K)$ and $\text{Gal}(L/K)$ act, respectively, on ...
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1answer
44 views

Give an example of a polynomial whose Galois group is isomorphic to $D_{10}$

I have been spending my leisure time determining the subfield lattices and corresponding Galois subgroup lattices of some splitting fields of polynomials. I have made the lattice diagrams for the ...
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1answer
65 views

Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
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Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
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1answer
35 views

$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group ...
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1answer
40 views

Trancendental extension Galois group

Let $K$ be a field and consider the extension $K(X)$ of rational functions with coefficients in $K$. It is common knowledge that $\text{Gal}(K(X)/K)$ is isomorphic to the group ${PL }_2(K)$, which is ...
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1answer
60 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
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1answer
26 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
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81 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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1answer
37 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
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1answer
28 views

Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?
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The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
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1answer
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Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ [duplicate]

Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ We must show that it is normal and separable. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it ...
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1answer
60 views

How many fields are there between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_{15})$?

$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that: $\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$ Is there a ...
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2answers
71 views

Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

Suppose $ \zeta$ is a primitive $ 11$-th root of unity and $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $ Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ Could someone please give me a hint ...
2
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1answer
35 views

Is the absolute Galois Group of $\Bbb Q$ countable?

Is $\text{Gal} (\overline{\Bbb Q}/\Bbb Q)$ countable or uncountable? It seems like it should be countable (because the algebraic closure of $\Bbb Q$ is countable and there are countably many ...
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Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
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Find the Subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$

As an exercise for myself I want to find the subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$. I know that it has Galois group $\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$, and this has ...
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1answer
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How to show that $\sqrt{5}$ is not in $Q(\sqrt{2},\sqrt{3})$

My strategy is that suppose $\sqrt{5}$ is in $Q(\sqrt{2},\sqrt{3})$, then it should be of the form $a+b\sqrt{2} +c\sqrt{3}+d\sqrt{6}$. After some tedious computation I can get a contradiction. Is ...
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Since field extensions are vector spaces over ground fields, how is linear algebra used to study them?

I've already taken a semester of Galois theory, so I'm not asking about anything that would be covered there, like the tower theorem. Specifically, multiplication by an element of a field extension is ...
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Galois group of splitting field over $\mathbb{Q}$

Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$ I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the ...
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40 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
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1answer
46 views

Finding Intermediate Fields of Field Extensions

Given $K$ the splitting field of $x^4 - x^2 - 1$ over $\mathbb{Q}$, we want to determine all intermediate fields $L$, $\mathbb{Q} \subset L \subset K$ and determine which are Galois. I've shown that ...
3
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2answers
42 views

$\alpha \alpha' \in K^2 $ if and only if $Gal(E/K) \cong C_2 \times C_2$

K= field of characteristic p $\neq$ 2, $c \in K-K^2$ and $F=K(\sqrt{c})$. Let $\alpha = a+b\sqrt{c}$ such that $\alpha \notin F^2$ and $E=F(\sqrt\alpha)$. Define $\alpha'=a-b\sqrt(c)$. How can I ...
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Structure of Galois group

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$ and $K|_F$ be the splitting field of $f(x)$. I want to prove the statement that if $G = \text{Gal}(K|_F)$ is cyclic then $[K:F] ...
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1answer
31 views

$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
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1answer
48 views

What are the roots of quintics?

I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general ...
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1answer
39 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
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1answer
40 views

Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
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1answer
32 views

Size of automorphism group of finite Galois extension

I have seen that if an extension is finite and Galois then the size of the automorphism group is equal to the degree of the extension - but is the converse always true? Ie if I have an extension and ...
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0answers
22 views

If $L=K(a_1,a_2,..,a_n)$ can we find an irreducible polynomial in $K[x]$ s.t $p(a_i)=0$?

I have the following question that I can't prove or find a counterexample for.Let $K$ a field and $L$ a finite filed extension of $K$ so that we can write $L=K(a_1,a_2,..,a_n)$ where all $a_i\in ...