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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Galois group permutation of roots

When considering the Galois group of the splitting field of the polynomial $x^3-2$, it is mentioned in my notes that $\sqrt[3]{2}$ can be mapped to $\sqrt[3]{2}$,$\sqrt[3]{2}\omega$ or ...
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973 views

Is the following field extension normal $\mathbb Q \sqrt {2+\sqrt 2}\mid \mathbb Q$

I would like to know if $\mathbb Q \left[\sqrt {2+\sqrt 2}\right ]: \mathbb Q$ is normal . The roots of the minimal polynomail is $\pm\sqrt {2\pm\sqrt 2}$ . Now the thing that i have really tried ...
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190 views

Does a tower of Galois extensions in $\mathbb{C}$ give an overall Galois extension?

If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension?
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1answer
102 views

Irreducible and separable polynomial

Let $f(x)$ be an irreducible polynomial over $F[X]$, with $char(F)=p$ prime number. We know that $ \exists t \in \mathbb{N} | f(x)=g((x^p)^t)$. We shall prove that $g(x)$ is irreducible and separable. ...
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194 views

Questions about the details of Abel's theorem

This is an extract of Wikipedia's page on Abel-Ruffini theorem The following proof is based on Galois theory. Historically, Ruffini and Abel's proofs precede Galois theory. One of the ...
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703 views

Subfields of the field of complex numbers with finite index rather than the real number field [duplicate]

Possible Duplicate: Finiteness of the Algebraic Closure For short, I wonder if there are other fields $F\subset \mathbb{C}$ rather than $\mathbb{R}$, with finite index $[\mathbb{C}:F]$. ...
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97 views

Rational roots existence proof

Problem statement We want to show that the polynomial $x^p-p\,2^p\,x+p^2\in\mathbb Z[x]$, $p$ prime, has no rational root. My approach We separate the proof in two steps: $p>2$ and $p=2$ ...
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1answer
111 views

Galois theory (Showing $G$ is not abelain)

Suppose $G$ is the Galois group of an irreducible degree $5$ polynomial $f \in \mathbb{Q}[x]$ such that $|G| = 10$. Then $G$ is non-abelian. Proof: Suppose $G$ is abelian. Let $M$ be the splitting ...
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1answer
31 views

If $K$ is a field extension of $F$ and $S \subseteq K$ is such that each $s \in S$ is $F$-algebraic, is it true that $F[S] = F(S)$?

If K is a field extension of F and $S\subseteq K$ is such that each s in S is F-algebraic, is it true that F[S] = F(S)?
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466 views

Tips for finding the Galois Group of a given polynomial

I am currently in an introductory Galois Theory course, and I thought it would be nice to compile a list of standard tricks for finding the Galois Groups of certain polynomials. I am studying from ...
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1answer
56 views

How many prime ideals are fixed by a given permutation?

Suppose $L$ is a finite Galois field extension of the rational number field $\mathbb{Q}$, and $B$ is the integral closure of $\mathbb{Z}$ in $L$. Let $\sigma$ be an element of the Galois group ...
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86 views

Irreducible polynomial $\frac{x^{n}+x^{m}-2}{x^{\gcd(n,m)}-1}$ over $\mathbb{Q}$.

I have to show that the polynomial $$f(x)=\frac{x^{n}+x^{m}-2}{x^{\gcd(n,m)}-1}$$ is irreducible over $\mathbb{Q}$, for all $n,m \in \mathbb{N}$. Any idea as to how I can show this.
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325 views

extension of automorphism of field to algebraically closed field

Suppose that F is a field contained in an algebraically closed field A. Prove that every automorphism of F can be extended to an automorphism of A.
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47 views

How would I show that $Z_7(t^9)$ subset $Z_7(t)$ isn't a normal extension where $Z_7(t)$ is a rational function field?

It's not clear to me what rational function fields are and the significance of normal extensions in the first place. $Z_7(t^9)$ seems smaller than $Z_7(t)$ so it makes sense one would be contained in ...
4
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1answer
105 views

Eisenstein generalization

Problem: Given a polynomial $f(x)=c_0+c_1\,x+\ldots+c_n\,x^n\in\mathbb{Z}[x]$, assume that exists $p\in\mathbb{Z}$ prime that satisfies: $p\,\nmid\,c_n$ $p\,\mid\,c_i,\;\forall i=0,\,\ldots,n-1$. ...
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1answer
627 views

Usage of finite fields or Galois fields in real world

I'm currently studying the theory of Galois fields. And I have a question, what practical usage of this finite fields? As stated in Wikipedia: Finite fields are important in number theory, ...
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129 views

two Non isomorphic root field-extension of the field.

Does there exist two non isomorphic minimal field extension ( root field) of $f= \frac {x^{64}-x}{x(x-1)} \in F_2[x]$ . I may be using wrong word here saying minimal field extension but in german ...
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1answer
82 views

$L/K$ field extension of degree $2^k$. Element $a$ in $L$ or $K$?

Let $L/K$ be a field extension of degree $2^k$, $k\in\mathbb{N}$. $a\in L$ and $f\in K[X]$ a polynomial of degree $d$ such that $f(a)=0$, $d$ odd. Show that $a \in K$. I know only the definition of a ...
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2answers
271 views

If $L/k$ and $\gcd(f(x),g(x))=h(x)$ in $k[x]$, then $\gcd(f(x),g(x))$ is also $h(x)$ in $L[x]$?

I am trying to prove this theorem: Let $L/k$ be field extension, and let $f(x),g(x)$ be two polynomials in $k[x]\setminus\{0\}$ and $h(x)=\gcd(f(x),g(x))$ in $k[x]$, then prove that the ...
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1answer
85 views

Inverse of the fundamental theorem of Galois Theory for finite extensions

Let $E/K$ be a finite Galois field extension. Then by the fundamental theorem of Galois theory, there is canonical bijection between the subgroups of $\mathrm{Gal}(E/K)$ and the intermediate field ...
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1answer
204 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
2
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1answer
315 views

What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
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146 views

etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
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1answer
185 views

Where is the calculation hiding in this proof about how $p$ splits in $\mathbb{Q}(\zeta_n)$?

I just worked through a proof in Daniel Marcus' book Number Fields that if $p\nmid n$, the inertial degree of any prime ideal of $\mathbb{Q}(\zeta_n)$ lying over $p$ is equal to the order of $p$ in ...
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1answer
302 views

An angle $\theta$ can be trisected if and only if $4x^3-3x+\cos\theta$ is reducible over $\mathbb{Q}(\cos\theta)$

I was given the above problem for homework. There is (what seems to be) a relevant proof in my textbook regarding the impossibility of trisecting $\pi/3$. In this proof, the identity $$\cos 3\theta = ...
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1answer
92 views

Real embedding of the splitting field of $X^3-2$

Does the splitting field of $X^3-2$ have a real embedding?
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52 views

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?.

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?. I don't know why is necessary a condition for $q$.
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1answer
482 views

Is the field extension $K(t):K$ normal? Is separable?

The field extension $K(t):K$ where $K$ is any field. I don't know how to apply definition of normal and separable in a transcendental case.
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1answer
208 views

searching the fixed field of an automorphism, and a primitive generator , in characteristic p.

Let $K$ be a field oh characteristic $p$. Let's take $\sigma \in \operatorname{Aut}(K(x),K)$ where $x$ is trascendental over $K$, where $\sigma(x)=x+1$. Find a primitive element of the fixed field of ...
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1answer
628 views

Computing the Galois group of $x^4+ax^2+b \in \mathbb{Q}[x] $

I want to compute the Galois group of some polynomials, but I want to see some examples first. For example this proposition could be helpful. I don't know how to prove it <.< Let's consider a ...
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794 views

Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$

I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a ...
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1answer
219 views

finite fields, a cubic extension on finite fields.

Let $q=p^n$ where $p$ is a prime number. For which $q$ is the extensión $\mathbb{F}_{q^3}/\mathbb{F}_q$ (i.e the extension of degree $3$) an extension of the form $\mathbb{F}_q(\root 3 \of \alpha )$ ...
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99 views

the proof of a proposition of galois theory

Let f(x) be a separable polynomial over the field K, with roots $r_1 , ... , r_n$ in it's splitting field F. Then f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of ...
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229 views

searching an explicit isomorphism of finite fields

Since all the finite field of $p^n$ elements are the splitting field of the separable polynomial $x^{p^n}-x$, all of them are isomphic. In particular if $f_1(x),f_2(x)$ are irreducible polynomials ...
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2answers
659 views

computing the galois group of a polynomial

Compute the Galois group of the splitting field of the polynomial $t^4-3t^2+4$ over $\mathbb{Q}$. I don't know how can I do this problem, the roots are very "ugly" maybe if I consider another basis ...
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0answers
132 views

splitting field extending an automorphism

Let $\varphi:K_1\to K_2$ be an isomorphism of fields. Let's consider a polynomial $p(x)\in K_1[x]$, and it's splitting field extension $ E_1/K_1 $, also the polynomial $\varphi(p(x))$ $\in K_2[x]$, ...
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How do we use Galois theory to show that an integral has no closed form? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? How do we use Galois theory to show that an integral has no closed form ? I know this is called ...
3
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1answer
147 views

What is the Galois group of $x^n + (x-1)^n $?

What is the Galois group of $x^n + (x-1)^n $ over the rationals in terms of the integer $n$ ? In case that is too hard , what is it for the first 20 integers ?
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1answer
101 views

What is the galois group of $x+3$ or $(x+1)(x+2)$ ? How about $A(x)B(x)$?

As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of ...
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1answer
120 views

Minimal polynomial of the form $\zeta_p+\frac{1}{\zeta_p}+\zeta_q+\frac{1}{\zeta_q }$?

We can calculate the minimal polynomial of $ 2cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over Q as x^3+x^-2x-1 and simlary for $2cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$. Now my question ...
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Determining a Galois group without factoring

Let $f \in \mathbb{Q}[X]$ be irreducible and let $L$ be its splitting field. Can something be said about the Galois group of $L$ over $\mathbb{Q}$ without computing the roots of $f$ in $\mathbb{C}$?
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1answer
426 views

A proof of the fundamental theorem of symmetric polynomials

I'm reading Exploratory Galois Theory by John Swallow. On page 123 he gives the following remark / alternate proof of the fundamental theorem of symmetric polynomials: Let $K$ be a field and $L$ ...
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1answer
388 views

Characteristic 2

Show that for a field $L$ of characteristic 2 there exist quadratic equations wich cannot be solved by adjoining square roots of elements in the field $L$. Attempt: In $\mathbb{Z_2}$ adjoining all ...
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2answers
471 views

Compositum of abelian Galois extensions is also?

Suppose I have a field $k$ and two extensions $K/k$ and $L/k$ which are both abelian Galois extensions of $k$. Then (assuming $K$ and $L$ are both contained in some bigger field) is the compositum ...
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284 views

Serre's Topics in Galois Theory

I am supposed to write a running notes on Hilbert Irreducibility Theorem from Serre's Topics in Galois Theory as part of my master's Algebra course work. But I don't have any knowledge of Algebraic ...
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56 views

A basis of Galois extention $L/K(R)$

Ler $R$ be a normal domain with its quotient field $K$. Let $L$ be a finite Galois extention of $K$. Let $T$ be the integral closure of $R$ in $L$. Then we can take a basis $t_1,\cdots t_m$ of vector ...
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Application of Galois theory to cubic polynomial

Given a cubic polynomial $f(x) \in \mathbb{Z}[x_{1},\dots,x_{n}]$. Suppose that $f(x)$ does not factor into three linear polynomials but contains a linear factor. Is the linear factor defined over ...
4
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1answer
242 views

A prime in a separable extension splits completly iff it does so in the galois closure.

I just did the following exercise out of Neukirch's Algebraic Number Theory: "A prime ideal p of K is totally split in the separable extension L|K iff it is totally split in the Galois closure N|K of ...
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Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$

Let $K$ be a field and $\mu_n$ a primitive $n$-th root of unity. Then I can embed $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$. For $K=\mathbb{Q}$ there would be even an ...
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1k views

Is every group a Galois group?

It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following ...