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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reducible polynomial modulo every prime

how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$. For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
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Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
17
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2answers
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How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
8
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1answer
691 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
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showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

I studied the cyclotomic extension using Fraleigh's text. To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, ...
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Constructing a Galois extension field with Galois group $S_n$

Constructing a Galois extension field $E$ with $Gal(E/F)= S_n$ How do I construct one?
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How to transform a general higher degree five or higher equation to normal form?

This is a continuation of How to solve fifth-degree equations by elliptic functions? How to transform a general higher degree five or higher equation to normal form? For xeample, a quintic equation ...
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2answers
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How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots \sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?

This is Exercise 18.14 from Algebra, Isaacs. $p_{1}\ ,\ p_{2}\ ,\ ... p_{n}$ are different prime numbers. How to show that $$\mathbb{Q}[\sqrt{p_{1}}, \sqrt{p_{2}}, \ldots \sqrt{p_{n}} ] = ...
27
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1answer
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Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
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5answers
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Galois group of $x^3 - 2 $ over $\mathbb Q$

I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with ...
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2answers
363 views

Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
6
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1answer
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Splitting fields of polynomials over finite fields

I can't follow a statement in my notes: "Let $K$ be a finite field, with $f \in K[X]$ an irreducible polynomial of degree $d$. Then any finite extension $L/K$ is normal, and so if $L$ contains one ...
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2answers
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Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8 Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...
4
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3answers
396 views

Minimal Polynomial of $\zeta+\zeta^{-1}$

Question is to find Minimal Polynomial of $\zeta+\zeta^{-1}\in \mathbb{Q}(\zeta)$ over $\mathbb{Q}$ Where $\zeta$ is primitive $13^{th}$ root of Unity. What all i know is that Minimal polynomial of ...
28
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3answers
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How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
24
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3answers
802 views

Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
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2answers
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“Standard” ways of telling if an irreducible quartic polynomial has Galois group C_4?

The following facts are standard: an irreducible quartic polynomial $p(x)$ can only have Galois groups $S_4, A_4, D_4, V_4, C_4$. Over a field of characteristic not equal to $2$, depending on whether ...
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3answers
1k views

What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
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4answers
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Do finite algebraically closed fields exist?

Let $K$ be an algebraically closed field ($\operatorname{char}K=p$). Denote $${\mathbb F}_{p^n}=\{x\in K\mid x^{p^n}-x=0\}.$$ It's easy to prove that ${\mathbb F}_{p^n}$ consists of exactly $p^n$ ...
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Why does an irreducible polynomial split into irreducible factors of equal degree over a Galois extension?

I've been struggling to prove this fact over the past day or so. Suppose $f(x)\in F[X]$ is irreducible over a field $F$ with $\deg(f)=n$, and let $L$ be the splitting field of $f(x)$ over $F$ with ...
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1answer
382 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
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4answers
408 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
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3answers
379 views

Rational Function Field over characteristic 0

Let $K=F(x)$ be the rational function field over a field $F$ of characteristic 0, let $L_1=F(x^2)$, and $L_2=F(x^2+x)$. How to show that $L_1\cap L_2 = F$?
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1answer
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Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
4
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2answers
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Constructing an explicit isomorphism between finite extensions of finite fields

Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal ...
2
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1answer
198 views

Transition between field representation

In a number of papers related to efficient implementation of the AES Sbox, people are computing stuff (the multiplicative inverse for instance) in GF(($2^4$)$^2$) instead of GF($2^8$). In some cases ...
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2answers
590 views

What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$?

Let $\zeta_n$ be the $n$-th primitive root of unity and $4 \mid n$. Consider the field extensions $\mathbb Q \subset \mathbb Q(\sin(2\pi k/n) \subset \mathbb Q(\zeta_n)$. What is the degree of the ...
4
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If Gal(K,Q) is abelian then |Gal(K,Q)|=n

Let $f(x)\in \mathbb Q[x]$ irreducible of degree $n$ and $K$ its splitting field over $\mathbb Q$. Prove that if $\operatorname{Gal}(K/\mathbb Q)$ is abelian, then $|\operatorname{Gal}(K/\mathbb ...
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1answer
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Minimal polynomial of intermediate extensions under Galois extensions.

Let $K$ be a Galois extension of $F$, and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, and $H=\mathrm{Gal}(K/F(a))$. Let $z_1, z_2,\ldots,z_r$ be left coset representatives of $H$ in $G$. Show that ...
5
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1answer
373 views

field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
2
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2answers
364 views

If $N$ is a normal subgroup of $G$ with $N$ and $G/N$ solvable, prove that $G$ is solvable

I'm trying to prove that if $N$ is a normal subgroup of $G$, with $N$ and $G/N$ solvable, then $G$ is solvable. Proving that $G/N$ is abelian would of course suffice, but I'm not sure if that's a ...
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The algebraic closure of a finite field and its Galois group

$F$ is an extension field of a field $K$. Let $F$ be an algebraic closure of $\mathbb{Z}_p $ ($p$ prime). Show that $(i)$ $F$ is algebraic Galois over $\mathbb{Z}_p$ $(ii)$ The map ...
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Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
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3answers
370 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
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3answers
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Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). ...
10
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1answer
203 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
8
votes
5answers
414 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
6
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1answer
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what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?

John Conway proved in his book, On Numbers and Games (ch6, theorem 49) that the set of all ordinals smaller than $\omega^{\omega^\omega}$ form a field of characteristic 2 that is isomorphic to ...
6
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2answers
393 views

Are quartic minimal polynomials over $\mathbb{Q}$ always reducible over $\mathbb{F}_p$?

This situation arose while studying biquadratic extensions. Let $\mathbb{Q}(\alpha)$ is some biquadratic extension, with $m(x)$ the minimal polynomial of $\alpha$. Suppose that ...
4
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1answer
492 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 ...
6
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1answer
379 views

Degree of field extension

If $p$ is odd prime and $c=\cos(\frac{2\pi}{p})$, $s=\sin(\frac{2\pi}{p})$ then for which values of $p$ does $\mathbb{Q}(s,c)=\mathbb{Q}(c)$?
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A Galois Group Problem

I couldn't figure out a proof of the following statement while I'm reading the book "Fields and Galois Theory" by J. Milne. Let $A$ to be a UFD and let $P$ be a prime ideal of $A$, and let ...
4
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1answer
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Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

If $\mathbb{F}_q$ is a finite field with $q=p^m$ elements where $p$ is prime, if $a\in\mathbb{F}_p^\times$, ¿Which are the irreducible factors of $f_a=x^q-x-a$ ? Attempt: If $\alpha$ is a root for ...
13
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2answers
938 views

Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize ...
4
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1answer
260 views

Resolvent of the Quintic…Functions of the roots

Last year Mathlover posted a very good question about Galois theory: specifically, the existence of funtions of roots which map to each other under permutations of those roots. You can see his ...
4
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2answers
287 views

What is the condition for a field to make the degree of its algebraic closure over it infinite?

As we all know, the algebraic closure often has an infinite degree. Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields. However, we may ...
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1answer
302 views

a property of the radical closure of a field

Definition 1: Let $K$ be a field. If $\alpha$ is an algebraic element over K, such that $\alpha^n \in K $ and such that $x^n-\alpha^n \in K[x] $is also irreducible over K. Then we call $ K(\alpha)$ a ...
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2answers
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Calculating the Galois group of an (irreducible) quintic

On my homework I have been asked to compute the Galois group of a quintic. I have no idea how to do this, except (a) I calculated that it was irreducible (brute-force) (b) Since it is irreducible, ...
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1answer
485 views

If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime

Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
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Calculate a galois group

I am trying to calculate the galois group $\operatorname{Gal}( \mathbb{Z}_q (\vartheta_p) : \mathbb{Z}_q) $, where $p$ and $q$ are different primes, $\mathbb{Z}_q$ $q$-adic ring, $\vartheta_p$ a ...