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 ...

learn more… | top users | synonyms

9
votes
2answers
2k views

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 ...
26
votes
2answers
2k views

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 ...
21
votes
2answers
1k views

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?
9
votes
1answer
859 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 ...
13
votes
4answers
2k views

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, ...
9
votes
3answers
483 views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= ...
6
votes
2answers
2k views

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?
9
votes
5answers
5k views

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 ...
5
votes
2answers
457 views

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 ...
15
votes
2answers
1k views

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}} ] = ...
31
votes
1answer
458 views

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 ...
6
votes
3answers
417 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 $L_1\cap L_2 = F$?
8
votes
2answers
661 views

When is $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
6
votes
1answer
1k views

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 ...
6
votes
1answer
1k views

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 $\Phi(n)$ ...
32
votes
3answers
8k views

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 ...
10
votes
2answers
2k views

Relationship between Cyclotomic and Quadratic fields

Since $\varphi(p)=p-1$ is even the p'th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field. What is this theorem ...
9
votes
2answers
2k views

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 ...
8
votes
3answers
2k views

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 ...
3
votes
2answers
399 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 ...
2
votes
1answer
203 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 ...
4
votes
2answers
190 views

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 ...
4
votes
3answers
476 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 ...
23
votes
2answers
2k views

“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 ...
10
votes
1answer
604 views

Constructive Proof of Kronecker-Weber?

This question is motivated by my attempt at solving Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ Consider ...
25
votes
3answers
866 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 ...
9
votes
4answers
1k views

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 $p^n$ ...
6
votes
3answers
2k 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.
8
votes
4answers
513 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 ...
7
votes
1answer
519 views

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 ...
6
votes
1answer
449 views

Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
7
votes
2answers
2k views

Galois Group of $X^p - 2$, p-an odd prime

Question is to Determine the Galois group of $x^p-2$ for an odd prime p. For finding Galois group, we look for the splitting field of $x^p-2$ which can be seen as $\mathbb{Q}(\sqrt[p]{2},\zeta)$ ...
4
votes
2answers
2k views

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 ...
3
votes
2answers
298 views

Quartic Equation having Galois Group as $S_4$

Suppose $f(x)\in \mathbb{Z}[x]$ be an irreducible Quartic polynomial with Galois Group as $S_4$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$.Now, the Question is: Prove that $K$ ...
-1
votes
2answers
625 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 ...
2
votes
3answers
651 views

Galois group and the Quaternion group

Let $E=\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $$ L = E \left( \sqrt{ ( \sqrt{2}+2 ) ( \sqrt{3} + 3)} \right) \ . $$ I want to show that $L/\mathbb{Q}$ is a Galois extension with the Quaternion group as ...
1
vote
1answer
168 views

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
votes
2answers
135 views

$f \in K[X]$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?

Let $K$ be a field and $f \in K[X]$ of degree $n$ with Galois group $S_n$. Let $a$ be a root of $f$, $L = K(a)$, and let $E$ be a intermediate field of the extension $K \subset L$. Prove that $E = K$ ...
5
votes
1answer
538 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
votes
2answers
444 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 ...
57
votes
3answers
962 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
31
votes
1answer
870 views

Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such ...
10
votes
3answers
1k views

Every finite abelian group is the Galois group of of some finite extension of the rationals

I'm trying to prove that every finite abelian group is the Galois group of of some finite extension of the rationals. I think I'm almost there. Given a finite abelian group $G$, I have constructed ...
12
votes
3answers
413 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 ...
26
votes
3answers
2k views

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. ...
11
votes
3answers
1k views

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
votes
1answer
222 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
456 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
votes
1answer
139 views

Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$

So first of all, I used the following from my lecture notes: If $f \in \mathbb{Z}[x]$ is primitive (gcd of all the coefficients is 1) - $f$ irreducible in $\mathbb{Z}[x] \Leftrightarrow$ f ...
6
votes
1answer
236 views

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 ...