2
votes
2answers
48 views

Galois Group for $x^5-1$

This question is an extension to the question in math.stackexchange.com/questions/759230/subfield-of-the-galois-group-of-x5-1 It seems the discussion in that topic is dead and I still have a major ...
1
vote
2answers
38 views

Galois Group Calculation

Calculate the Galois Group $G$ of $K$ over $F$ when $F=\mathbb{Q}$ and $K=\mathbb{Q}\big(i,\sqrt2,\sqrt3 \big)$. My thoughts are as follows: By the Tower Lemma, we can see that ...
1
vote
0answers
20 views

Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
2
votes
1answer
57 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
0
votes
1answer
17 views

Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...
0
votes
1answer
18 views

In algebraic extension, field homomorphism induces isomorphism.

I read this page's first answer. But I'm curious about why $\varphi$ induces injective map $S \to S$. Isn't it possible to make $\varphi(\alpha)= k$ such that $k$ is not root of $f(X)$?
1
vote
1answer
49 views

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$?

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$? I think that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}):\mathbb{Q}] = 8$, but not really sure how to ...
1
vote
1answer
56 views

Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
2
votes
3answers
72 views

Is is true that $\zeta$ has finite order?

Let $\zeta$ be a complex number on the unit circle $\{z\in \mathbb{C}: |z|=1\}$.Suppose that $[\mathbb{Q}(\zeta):\mathbb{Q}] < \infty$.Is it true that $\zeta ^n=1$ for some positive integer $n$?
5
votes
1answer
61 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
1
vote
1answer
27 views

Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
3
votes
2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
5
votes
2answers
64 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
3
votes
1answer
35 views

The Galois closure

If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
1
vote
1answer
31 views

Proving the Galois Group of an extension is abelian

Let $E_{1}, E_{2}$ be subfields of $\mathbb{C}$. Suppose $E_{1}|\mathbb{Q}$ and $E_{2}|\mathbb{Q}$ are finite Galois extensions and $G(E_{1}:\mathbb{Q})\cong$ $\mathbb{Z}_{6}\cong$ ...
3
votes
1answer
58 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
0
votes
1answer
40 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
2
votes
1answer
35 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
0
votes
1answer
39 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
0
votes
1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
2
votes
2answers
46 views

Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
-1
votes
1answer
16 views

Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
1
vote
1answer
30 views

Every Intermediate Field of Abelian Galois Field Extension is Splitting Field of a Separable Polynomial

This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
1
vote
2answers
40 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
5
votes
0answers
56 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
5
votes
0answers
87 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
1
vote
1answer
65 views

How do we explain the existence of complex conjugation?

We can define the complex numbers by writing $\mathbb{C} = \mathbb{R}[i]/(i^2+1),$ where $\mathbb{R}$ is to be regarded as a commutative ring. Furthermore, since $\mathbb{R}$ happens to be a field, ...
3
votes
1answer
28 views

Can't find intermediate subfield of these cyclotomic extensions!

I'm to look at $\mathbb{Q}(\zeta_7)$ and $\mathbb{Q}(\zeta_{10})$, and they both have a common thing I can't see how to work around. For $\mathbb{Q}(\zeta_7)$, skipping some details I doubt matter ...
2
votes
1answer
45 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields ...
2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
0
votes
0answers
47 views

Galois Theory: An automorphism fixes a field if and only if it fixes the set of generators.

Let $F/K$ be a field extension. Let $a_{1},...,a_{n}\in{F}$ and $E:=K(a_{1},...,a_{n})$. Then how do we show $\sigma\in{Aut_{E}F}$ if and only if $\sigma(a_{i})=a_{i}$ for all $i=1,2,...,n$? Any ...
1
vote
1answer
38 views

The fixed field of a galois group if the characteristic is $p$.

Consider the following proposition with its relative proof: Let $k$ be an algebraically closed field of characteristic $0$. a) If $L$ is a subfield of $k$, then every elements of $\operatorname ...
0
votes
0answers
38 views

normal extension of $\mathbb{Q}$

We define an algebraic extension $K/F$ to be normal if every irreducible $f \in F[x]$ with one root in $K$ splits in $K[x]$. Now, in my lectures it was stated that $\mathbb{Q}(i)$ is a normal ...
4
votes
1answer
105 views

What are the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$?

I'm trying to find the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$. This is easily seen to be a degree $14$ extension of $\mathbb{Q}$. I found that there is a unique subfield of degree $2$ over ...
1
vote
1answer
35 views

Simple radical extension

Let $F/K$ be a Galois (finite) extension with solvable group. Must $F$ be a simple radical extension of $K$? or at least have an intermediate field which is a simple radical extension? If $F/K$ is ...
2
votes
0answers
20 views

Galois group of composite of Galois extensions

I'm reading through the proof in Dummit and Foote p. 593 that $$\operatorname{Gal}(K_1K_2/F) \cong H := \{(\sigma, \tau) \in \operatorname{Gal}(K_1/F) \times \operatorname{Gal}(K_2/F) \mid ...
1
vote
1answer
45 views

Why is the Galois closure of $K/F$ the composite of the Galois conjugates of $K$?

Suppose $K/F$ is a field extension with Galois closure $L$, and let $G=\operatorname{Gal}(L/F)$. Why is $L$ the same as the composite of the Galois conjugates $\sigma(K)$ for $\sigma\in G$? I know ...
0
votes
1answer
31 views

Degree of extensions and their composite

Let $F<E$ and $F<K$ be finite extensions and assume that $EK$ is defined as composite of two fields. I need to show that $[EK:F] \leq [E:F][K:F]$, with equality if $[E:F]$ and $[K:F]$ are ...
0
votes
0answers
27 views

Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
1
vote
0answers
25 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
5
votes
0answers
129 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
2
votes
1answer
104 views

How to show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$

So as title says I wanna show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ So I know that the minimal polynomial of $\mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ is $x^{4}-6x^{2}+7$ ...
0
votes
0answers
48 views

Galois group of a particular polynomial

What is the Galois group of the polynomial $X^n − 3$ over $\mathbb Q$? (Here $n$ is greater than $2$.)
2
votes
2answers
53 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
2
votes
1answer
47 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
3
votes
1answer
35 views

A polynomial with solvable Galois group and solution by radicals [duplicate]

Suppose $f(x)\in \mathbb{Q}[x]$ has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of $f(x)$ in terms of ...
0
votes
2answers
65 views

Prove that $\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=2^{n}$ [duplicate]

Let $p_{1},p_{2},\ldots,p_{n}$ be $n$ primes,$\left(p_{i},p_{j}\right)=1$ if $i\neq j$ . Prove that ...
0
votes
0answers
38 views

Prime polynomials over GF(q)

Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
1
vote
2answers
42 views

Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
3
votes
0answers
45 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...