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 (galois-...

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2
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2answers
62 views

Question in finding a new $\mathbb{Q}$-basis for $F/\mathbb{Q}$.

Let $F$ be the splitting field of $x^4 - 2$ over $\mathbb{Q}$. Let $G$ be its Galois group. When viewed as a $\mathbb{Q}$- vectorspace, $F$ has the following basis: $$\mathcal{B}=\{1,2^{1/4},2^{1/2},...
4
votes
0answers
25 views

Is an algebraic field extension $k \subseteq K$ normal if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$?

Over a perfect field $k$ it is well known that an algebraic field extension $k \subseteq K$ is normal if and only if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$, as ...
0
votes
1answer
15 views

irreducibility of minimal polynomial

Suppose degree of minimal polynomial of $\alpha$ over $F$ is relatively prime to $\beta$ over $F$ then prove that minimal polynomial of $\beta$ over $F$ is irreducible over $F(\alpha)$. I suppose it ...
1
vote
2answers
39 views

Let $E\supseteq F$ be a splitting field of $f(x)=x^6+1\in F[x]$. Find $[E:F]$.

Let $E\supseteq F$ be a splitting field of $f(x)=x^6+1\in F[x]$. Find $[E:F]$ for $F=\mathbb{Z}_2$ and $F=\mathbb{Q}$. I think in $\mathbb{Z}_2$, we can rewrite it as $f(x)=x^6-1=(x^3-1)(x^3+1)=(x^3-...
0
votes
1answer
19 views

$E/F$ is a finite Galois extension. Let $b\in E$, and $b_1=b, b_2…$ are the orbit of b under the action

Let $E/F$ be a finite Galois extension. So $E=F(a)$. Let $b\in E$, and let $b_1=b, b_2,...,b_n$ be the orbit of $b$ under the action of the Galois group $G$. (a)Show that the minimal polynomial of $...
1
vote
0answers
22 views

Proof verification: Show that the fixed field is $\mathbb{Q}(\sqrt{3})$

Let $H$ be the subgroup $\{i,\alpha\}$ of $\text{Gal}_{\mathbb{Q}}\mathbb{Q}(\sqrt{3},\sqrt{5}),$ where $i$ is the identity map and $\alpha$ is defined as $\alpha(\sqrt{3})=\sqrt{3}$,$\alpha(\sqrt{5})=...
4
votes
1answer
40 views

Example of field's normal closure that's not Abelian?

Suppose $K$ is a global field, $L/K$ is a field extension, and $M$ = normal closure of $L$ (over $K$). Is it possible that Gal($M/L$) is not Abelian? In all cases I know, $L$ is formed from $K$ by ...
1
vote
1answer
24 views

Totally real Galois extension of given degree

Let $n≥1$ be an integer. I would like to prove (or disprove) the existence of a subfield $K \subset \Bbb R$ such that $K/\Bbb Q$ is Galois and has degree $n$. It is easy to construct such a subfield ...
0
votes
2answers
35 views

Show that a transcendental simple extension has infinitely many intermediate fields. [on hold]

Show that a transcendental simple extension has infinitely many intermediate fields. I've been working on this for awhile but can't figure it out. Help would be greatly appreciated, Thanks!
0
votes
3answers
85 views

Show that $\mathbb{Q}(b)$ where $b=2^{1/3}+\zeta_3$ is equal to $\mathbb{Q}(2^{1/3}, \zeta_3)$ [on hold]

Show that $\mathbb{Q}(b)$ where $b=2^{1/3}+\zeta_3$ and $\zeta_3$ is a third root of unity is equal to $\mathbb{Q}(2^{1/3}, \zeta_3)$? I am not sure how to get $2^{1/3}$ and $\zeta_3$ from $b$... ...
4
votes
2answers
85 views

Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
0
votes
1answer
22 views

Let f in F[x] be irreducible, and let E/F be Galois. Factor f in E[x] to get lower degrees.

Let $f \in F[x]$ be irreducible, and let $E/F$ be Galois. Then $f$ might factor in $E[x]$ into irreducible factors of smaller degree. Show that all of these have the same degree. I have a rough idea....
2
votes
2answers
84 views

Calculating Galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb{Q}$

What's a quick elegant way to compute the galois group of $\mathbb Q(\cos \frac \pi 8)/\mathbb Q$? I found the minimal polynomial to be $x^4-x^2-\frac 18$ but computing things directly is just ...
6
votes
1answer
36 views

Basic application of fundamental galois theorem?

Let $E/F$ be a galois extension with group $S_n$. let $G$ be the stabilizer of $1$ and $H$ be generated by the cycle $(1,\dots ,n)$. I need to find $[E^GE^H:F],[E^G:F],[E^H:F],[E^G\cap E^H:F]$. For $[...
4
votes
3answers
57 views

Slick proof $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}p, \sqrt[\leftroot{-2}\uproot{2}p]{2}]/\mathbb Q)\cong S_p$?

After having seen a lengthy and painful calculation showing $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}3, \sqrt[\leftroot{-2}\uproot{2}3]{2}]/\mathbb Q)\cong S_3$, I'm wondering whether there's a ...
2
votes
1answer
57 views

extending rational functions over finite fields

This is probably similar to the question in the link, but i'm not sure how to solve it either.. I want to prove $\mathbb F_p(t)/\mathbb F_p(t^p-t)$ is Galois, compute its Galois group, and describe ...
3
votes
1answer
31 views

Equivalent condition for $F[a^{\frac 1p}]=F[b^{\frac 1p}]$ for $\mathrm{char}F$ coprime to $p$

Let $p$ be prime and suppose $\mathrm{char}F$ is coprime to $p$ and $F$ contains the roots of unity. Why is it true that $F[a^{\frac 1p}]=F[b^{\frac 1p}]$ if and only if there's so $i$ coprime to $p$ ...
1
vote
1answer
41 views

Proving an extension is galois and describe its automorphisms

I have the extension $\mathbb F_3[A]/\mathbb F_3$ where the $A$ are the roots of $x^{80}-1$. I need to prove this extension is Galois, find the Galois group, and describe the automorphisms. but I'm ...
3
votes
1answer
59 views

prove $[E:K\cap L]=[E:K][E:L]$

Let $E/K,E/L$ be finite Galois extensions with Galois groups $G,H$. If $G\cap H= \left\{1\right\}$, show $G\cdot H = \left\{ g h \mid g\in G,h\in H \right\}$ is a group iff $[E:K\cap L]=[E:K][E:L]$. ...
1
vote
2answers
20 views

Is the following polynomial solvable in radicals?

Is the polynomial $x^8-x^6+2x^4-6x^2+1$ solvable in radicals over $\mathbb{Q}$? I am unsure how to solve this. I don't know how to compute the Galois group, and the discriminant seems much to hard to ...
3
votes
0answers
54 views

Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
3
votes
1answer
59 views

Is the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + t^2)$ irreducible over $k(t)$?

Let $k$ be an algebraically closed field of characteristic 2 and let $k(t)$ be rational function field of one variable. Consider the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + ...
0
votes
1answer
17 views

Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $ \mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $ x^5 - x + 1$. If we let $\zeta$ be a root of this ...
5
votes
1answer
93 views

Galois group of the $x^8+8\in\Bbb Q[x]$

I have this problem because I think the extension has degree $16$ but I can't decide the group: I think it could be $\Bbb Z_4\times\Bbb Z_2\times\Bbb Z_2$ but not surely: First I do complex roots $$x^...
2
votes
1answer
32 views

minimal polynomial in Kummer extension

Let $n>1$ be an integer. Let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Let $\mu_n\subseteq K$ be the set of $n$-th roots of ...
2
votes
2answers
35 views

Embeddings of pure cubic field in complex field

I know that the complex embeddings (purely real included) for quadratic field $\mathbb{Q}[\sqrt{m}]$ where $m$ is square free integer, are $a+b\sqrt{m} \mapsto a+b\sqrt{m}$ $a+b\sqrt{m} \mapsto a-b\...
11
votes
0answers
67 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
6
votes
2answers
78 views

Structure and normality of the Galois group of $x^{15}-15 \in \mathbb{Q}[x]$.

Let $f(x) = x^{15}-15\in \mathbb{Q}[x]$. By Eisenstein's Criterion (using 3 or 5), $f$ is irreducible. Then $L=\mathbb{Q}(\sqrt[15]{15}, \omega)$ is the splitting field of $f$, where $\omega$ is a ...
1
vote
2answers
81 views

Irreducible polynomial in finite-fields

Let $\overline{\mathbb{F}}_2$ be an algebraic closure of $\mathbb{F}_2$, and let $\alpha\in\overline{\mathbb{F}}_2$ be such that $\alpha^2+\alpha+1=0$. Prove that if a polynomial $P$ is irreducible in ...
0
votes
1answer
44 views

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit?

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit? I know the definition of orbit and I know that $\sigma$ in Galois group changes roots. I ...
4
votes
2answers
52 views

When is $\overline{K}/K$ a Galois extension of $K$?

When is $\overline{K}/K$ a Galois extension of $K$, where $\overline{K}$ stands for the algebraic closure of $K$? I have the following three extensions: $\overline{\mathbb{Q}}/\mathbb{Q}$,$\...
5
votes
2answers
94 views

If $f$ has more than one root in $K$, then $f$ splits and $K/k$ is Galois?

Let $f \in k[x]$ be an irreducible polynomial of prime degree $p$ such that $K \cong k[x]/f(x)$ is a separable extension. How do I see that if $f$ has more than one root in $K$, then $f$ splits and $K/...
2
votes
1answer
37 views

When do the lifting properties characterize covering spaces?

A covering map $p:E \rightarrow B$ is continuous map satisfying the following: For every point $b\in B$, there exists a neighborhood $U$ of $b$, such that $p^{-1}(U)$ is a disjoint union of open ...
1
vote
0answers
26 views

Closure of a subgroup in an infinite Galois extension

Let $K/F$ be a Galois extension. Let $H$ be a subgroup of $G=\mathrm{Gal}(K|F)$. Let $\mathcal{N}=\lbrace N\subseteq G\text{ }|\text{ } N= \text{Gal}(K|E)\text{ where }[E:F]<\infty \text{ and } E/...
1
vote
0answers
23 views

Radical-solvable extensions

I'm studying Field and Galois Theory with different books and now I have a doubt about what is the exact statement of Galois' theorem. Some books define radical and solvable extensions but other books ...
2
votes
1answer
41 views

Determine possible degree of monic irreducible polynomial given degree of Galois extension.

Suppose $L/K$ is a separable normal field extension with $[L : K] = 21$. (i) What integers n are possibly equal to the degree of a monic irreducible polynomial $f(x) ∈ K[x]$ for which $L/K$ is a ...
1
vote
1answer
55 views

Irreducible polynomials over $\mathbb{Z}_2[x]$

Prove that the polynomial $1+x+..+x^m$ is irreducible over $\mathbb{Z}_2$ if and only if $m+1$ is a prime number and 2 is a primitive root in $\mathbb{Z}_{m+1}$ Is there any proof without using ...
-1
votes
0answers
29 views

galois group of finite extension of finite field is abelian

How can I prove "the Galois group of a finite ext. of a finite field is abelian"? give me some example.
1
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0answers
24 views

Galois group of a general monic septic or octic polynomial.

How to determine Galois group of a general monic septic or octic polynomial.Could anyone suggest any reference material like some research paper or book. Edit:Even a condition to decide the ...
6
votes
1answer
53 views

Is $\mathbb{R}$ a finite field extension? [duplicate]

Is there a field $K \subseteq \mathbb{R}$ with $[\mathbb{R} : K] < \infty$? My intuition tells me no: I would imagine that $K(\alpha)$ would be missing some $n$th root of $\alpha$ for all $x \in \...
7
votes
1answer
124 views

Galois group of $x^5-5x+10$

I was illustrating the theorem on solvability by radicals through some examples of degree $5$ polynomials. One I chose was $x^5-5x+10$. I was (perhaps wrongly) going to prove that the Galos group is $...
4
votes
5answers
68 views

Galois extension definition.

Let $L,K$ be fields with $L/K$ a field extension. We say $L/K$ is a Galois extension if $L/K$ is normal and separable. I don't fully understand this definition, is it saying that 1) $L$ has to be ...
16
votes
1answer
169 views

Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?

Assume that $p(x)\in \Bbb{Q}[x]$ is irreducible of degree $n\ge3$. Is it possible that $p(x)$ has three distinct zeros $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1-\alpha_2=\alpha_2-\alpha_3$? ...
5
votes
1answer
97 views

Degree of the difference of two roots

Let $f\in \mathbb Q[x]$ be an irreducible monic polynomial of degree $n$ and let $\alpha,\beta\in \overline{\mathbb Q}$ be two distinct roots of $f$. Is it possible to find a lower bound on the degree ...
4
votes
1answer
61 views

Bound for the degree

Let $K$ be a perfect field and let $f\in K[x]$ be a monic irreducible polynomial of degree $n$. Denote by $\alpha,\beta$ two distinct roots of $f$. Is the following bound true? $$ [K(\alpha-\beta):K]\...
-1
votes
1answer
11 views

$L$ and $F$ are separable extension of $K$ and stay in the same bigger field $M$. Prove that $LF$ is separable extension of $K$.

$L$ and $F$ are separable extension of $K$ and stay in the same bigger field $M$. Prove that $LF$ is separable extension of $K$. I see the solution of the finite case but in the infinite I just think ...
4
votes
1answer
35 views

Cyclic Galois group of even order and the discriminant

I am stuck on the following problem: Let K be a field of characteristic $\neq 2$ and $f\in K[X]$ a separable irreducible polynomial with roots $\alpha_1,\ldots \alpha_n$ in a splitting field $...
7
votes
0answers
119 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
1
vote
2answers
17 views

Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$

Let $K_0$ denote the prime subfield of $K$ then $\text{Gal}(K/K_0)=\text{Aut}(K)$. This is a statement in the book I am looking at, it is given without proof or further explanation. I don't see ...
1
vote
3answers
54 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...