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

0
votes
1answer
42 views

Does the succesion of two radical extensions yield a radical extension only in the obvious case?

Answering that recent stackoverflow question, I encountered the following related problem : Let $n,m,p\geq 2$ be integers, and let $K$ be a subfield of $\mathbb C$ containing all $nmp$-th roots of ...
2
votes
0answers
143 views

Galois false solution to the quintic equation

I am looking for the false solution Galois gave to the quintic equation before discovering group theory.
2
votes
1answer
46 views

is the field extension $\mathbb{Z}_{3}(X)\supseteq \mathbb{Z}_{3}(X^2)$ normal

Is this field extension: $\mathbb{Z}_{3}(X)\supseteq \mathbb{Z}_{3}(X^2)$ normal? I pick $t^2-X^2\in \mathbb{Z}_{3}(X^2)$ which is irreducible over $\mathbb{Z}_{3}(X^2)$ as it's zeros are $t=\pm ...
6
votes
2answers
572 views

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
2
votes
0answers
35 views

Parametric quadratic equation in Q(a)

Are there any algebraic numbers $a$ with the following property: for all $q\in\mathbb{Q}$, the square roots of $a+q$ are in $\mathbb{Q}(a)$ ? I`m fairly certain that the answer is no, but how does ...
2
votes
1answer
103 views

Abstract Algebra: Field extensions

I'm trying to prove the following but it causes me a lot of trouble: Let $L$ be a finite extension of degree $n$ of a field $K$ with characteristic $0$. Let $\sigma_1,\dots,\sigma_n$ be different $K$ ...
3
votes
1answer
73 views

Basic proof Galois Theory

i was looking a proof of the following: Let $k\subset F, F\subset K$ be Galois field extensions, then $Gal (K/F)$ is a normal subgroup of $Gal(K/k)$. I understand the proof but it start using the ...
10
votes
1answer
284 views

Old vs. Modern Galois theory

The original Galois theory was developed to answer the question of the expressibility of the roots of polynomial equations with arithmetic operations and radicals. However it seems that later ...
5
votes
1answer
335 views

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 ...
3
votes
1answer
91 views

Radical extension over $\mathbb{Q}$

Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d$. Prove that $E=\mathbb{Q}(\sqrt[d]{a})$. ...
12
votes
1answer
242 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
0
votes
1answer
760 views

Field extension, primitive root of unity

Let $\xi_5$ be a primitive fifth root of unity in $\mathbb{C}$, then we know the extension $\mathbb{Q}(\xi_5)\supseteq\mathbb{Q}$ has degree 4 so the Galois group is of order 4. I am trying to find ...
0
votes
2answers
75 views

field extension and equivalent automorphisms

Suppose we have a field extension $L\supseteq K$ and two automorphisms $\sigma,\tau\in Gal(L:K)$ that are equal for every generator $\alpha_{i}$ (i.e $\sigma(\alpha_{i})=\tau(\alpha_{i}),\forall i)$ ...
3
votes
2answers
117 views

What is the minimal polynomial of $x$ over $k(x^p - x)$?

Let $k=\Bbb F_p$, and let $k(x)$ be the rational function field in one variable over $k$. Define $\phi:k(x)\to k(x)$ by $\phi(x)=x+1$. I know that $\phi$ has order $p$ in $\operatorname{Gal}(k(x)/k)$ ...
1
vote
0answers
43 views

Explicit display of the contraction of an ideal in polynomial ring extensions

$K$ is a finite Galois extension of $k$, $I=(f_1,\dots,f_m)$ an prime ideal in $K[X_1,\dots,X_n]$, then what is $I\cap k[X_1,\dots,X_n]$? Is it ...
5
votes
1answer
212 views

Kummer extensions

I want to prove the following: If $F$ contains all $n$-roots of unity and $\operatorname{char}F$ not divides $n$ then $K:=F(\sqrt[n]{a_1},\sqrt[n]{a_2},\ldots,\sqrt[n]{a_m})/F$ is a Galois abelian ...
1
vote
1answer
153 views

extending an isomorphism of a field to an embedding

Let $K$ be an algebraic extension of $F$, contained in an algebraic closure $E$ of $F$. Let $\alpha \in K$, $m(x)$ be its minimal polynomial over $F$ and $\beta$ a root of $m(x)$ in $E$, then there ...
1
vote
1answer
134 views

adjoining root of an irreducible polynomial.

It's been a while since I last touched Galois theory. I want to check if this theorem is true: Let $f$ be an irreducible polynomial over $\mathbb{Q}$ and $\varphi$ be a root of $f$. Then $f$ is ...
3
votes
1answer
396 views

Action of a Galois Group on an Algebraic Variety

I've to solve the following exercise. Let's be $X$ a connected algebraic variety over $k$ and let $K$ be a finite Galois Extension of $k$ with Gaolis Group $G$. Now I have to prove that $G$ acts ...
1
vote
2answers
226 views

$F$ field, $\alpha$ separable on $F$. Is $F(\alpha)$ a separable extension of $F$?

Let $F$ be a field, and let $\alpha$ be algebraic and separable over $F$. Is $F(\alpha)$ a separable extension of $F$? By "$\alpha$ is separable" I mean that its minimum polynomial over $F$ is ...
5
votes
1answer
758 views

find a splitting field of $x^4+x^2-1$ over $\mathbb{Q}$

I am trying to find a splitting field and its degree of $x^4+x^2-1$ over $\mathbb{Q}$ I make a substitution $x^2=u$ then I get $u^2+u-1=0$ and get 4 solutions ...
3
votes
1answer
150 views

Describing the impossibility of trisecting the angle to high school students.

Does anyone have an idea on whether it would be possible to present the proof of the impossibility of trisecting the angle (or doubling the cube, for example) in order to demonstrate the power of ...
2
votes
1answer
210 views

Finding inverse in the field extension

I need a hint to solve a problem in the algebra book of Dummit and Foote. It is what is the inverse of $\dfrac{1+\theta}{1+\theta+\theta^{2}}$ in which $\theta$ is the root of $x^3-2x-2=0$. I think ...
10
votes
1answer
2k views

Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
3
votes
2answers
3k views

find a degree and splitting field for $x^4-2$ over $\mathbb{Q}(i)$

let $K=\mathbb{Q}(i)$ and let $f=x^4-2$. Find the splitting field, its degree and the basis. My solution First I find roots of the polynomial $x_{1,2}=\pm\sqrt[4]{2},\hspace{2mm}x_{3,4}=\pm i ...
2
votes
0answers
158 views

Automorphism of Field Extension Statement

In our Galois Theory course the following statement was given to us in lectures: Suppose that $L : K$ is finite and normal, and $α, β ∈ L$ are roots of an irreducible polynomial $m ∈ K[x]$. Then ...
1
vote
0answers
106 views

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of ...
1
vote
1answer
87 views

Computing $[\mathbb{Q}(\sqrt{6}):\mathbb{Q}]$

If $p$ is a prime, the polynomial $X^n-p$ is irreducible over $\mathbb{Q}$, so $\sqrt[n]{p} $ is algebraic over $\mathbb{Q}$. Hence $[\mathbb{Q}(\sqrt[n]{p}):\mathbb{Q}]$. But $p$ is not a prime, ...
1
vote
1answer
60 views

$\left[LF:K\right]\leq\left[L:K\right]\left[F:K\right]$

Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field. Prove that $\left[LF:K\right]\leq\left[L:K\right]\left[F:K\right]$ If $L,F$ are finite extensions over $K$, I'm ...
3
votes
1answer
45 views

Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field (cont)

Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field. Prove that $\left[LF:K\right]\leq\left[L:K\right]\left[F:K\right]$ Help me. Thanks a lot.
0
votes
1answer
39 views

Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field.

Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field. Prove that if $L=K(S)$, with $S$ is a nonempty subset of $L$ then $LF=F(S)$. Thank for any insight.
1
vote
1answer
157 views

Composite of two algebraic extensions is algebraic.

Let $L,F$ be extensions of the field $K$ and are contained in a common field. Prove that, if $L$ and $F$ are algebraic extensions over $K$ then $LF$ is also a algebraic extension over $K$. Help me a ...
1
vote
1answer
275 views

On irreducible polynomial over normal extension

Let $L/K$ be a normal extension and a irreducible polynomial $f(X) \in K[X]$. Prove that, if $f$ is reducible over $L$ then $f$ is factored into product of irreducible factors with same degree. ...
1
vote
2answers
312 views

Cubic with repeated roots has a linear factor

If $f$ is a cubic polynomial with a repeated root over a field then $f$ has a linear factor. I think that is true in perfect fields but I don't know how to prove it.
1
vote
1answer
82 views

A question about cubic roots of rational numbers

I'm trying to understand if, given $K$ a cubic cyclic extension of $\mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a third primitive root of unity, it always exists $b \in \mathbb{Q}$ such that $\sqrt[3]{b} ...
6
votes
1answer
126 views

Galois group of irreducible polynomial in a field of characteristic zero in which every element is a perfect square

This is one of the exercises during my reading of Ian Stewart's Galois Theory. Whether the following statement is true: If $K$ is a field of characteristic zero in which every element is a ...
1
vote
2answers
361 views

On polynomial of prime degree.

Let $K$ be a field, $f(X)\in K[X]$ be a polynomial of prime degree. Assume that for all extension $L$ of $K$, if $f$ has roots in $L$ then $f$ splits over $L$. Prove that either $f$ is irreducible ...
0
votes
1answer
102 views

Composite of two purely inseparable extensions is purely inseparable.

Let $L,F$ be extensions of the field $K$ and are contained in a common field. Prove that, if $L$ and $F$ are purely inseparable extensions over $K$ then $LF$ is also a purely inseparable extension ...
7
votes
1answer
143 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
0
votes
1answer
145 views

Let $p$ be a prime. Compute the Galois group of the polynomial $f(X)=X^p-1$ over $\mathbb{Q}$.

Let $p$ be a prime. Compute the Galois group of the polynomial $f(X)=X^p-1$ over $\mathbb{Q}$. A result I known: Let $K$ be a field with characteristic $0$ and $L$ be the splitting field of that ...
2
votes
0answers
29 views

Minimal polynomial of Galois extension [duplicate]

Let $K$ be a Galois extension of $F$ and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$ and $H=Gal(K/F(a))$. Let $\tau_{1},\dots,\tau_{r}$ be left coset representatives of $H$ in $G$. Show that $\min(F,a) ...
9
votes
1answer
242 views

Intermediate field, normal closure and Galois group

Let $K/F$ be Galois with $G=Gal(K/F)$ and let $L$ be an intermediate field. Let $N\subseteq K$ be the normal closure of $L/F$. If $H=Gal(K/L)$ show that $Gal(K/N)=\cap_{\sigma\in G}\sigma ...
1
vote
1answer
80 views

Necessary and sufficient condition under which a Galois extension is cyclic

I want to know a necessary and sufficient condition under which a Galois extension $L$ over $K$ is cyclic. I mean the Galois group of $L$ over $K$ is cyclic.
4
votes
1answer
78 views

Let $L/K$ be a Galois extension with $Gal(L/K)=A_{4}$. Prove that there is no intermediate subfield $M$ of $L/K$ such that $[M:K]=2$.

Let $L/K$ be a Galois extension with $Gal(L/K)=A_{4}$. Prove that there is no intermediate subfield $M$ of $L/K$ such that $[M:K]=2$. Please tell me a hint. Thanks a lot.
6
votes
0answers
307 views

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$ Corollary $3.20$ page $267$ of Hungerford - Algebra: "Every proper ...
3
votes
0answers
75 views

Parametric 12-deg and 14-deg equations with group $PGL(2,11)$ and $PGL(2,13)$?

We have, $$x^{12} - a x^{11} - 33x^8 + 22a x^7 - 11a^2 x^6 + 363x^4 - 121a x^3 + 121a^2x^2 - 23a^3x - 11^3 + a^4=0$$ $$x^{12} - a x^{11} - 11a x^9 - 44a x^7 - 88a x^5 - 88a x^3 - 32a x - a^2=0$$ ...
1
vote
1answer
227 views

Compositum of finite field extensions

If $L_{1}$ and $L_{2}$ are field extensions of $F$ that are contained in a common field, show that $L_{1}L_{2}$ is a finite extension of $F$ if and only if both $L_{1}$ and $L_{2}$ are finite ...
22
votes
0answers
345 views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
2
votes
0answers
124 views

Exercise $7.7$ page $82$ Ian Stewart, Galois Theory

Prove that an angle $\theta$ can be trisected by ruler and compasses iff the polynomial $4t^{3}-3t-\cos\theta$ is reducible over $\mathbb{Q}\left(\cos\theta\right)$
0
votes
1answer
60 views

On algebraically closed field

Prove that $K$ is a algebraically closed field iff there are not exist algebraic extensions over $K$ of degree $>1$ Can anyone tell me a hint to solve the problem?