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
0answers
4 views

Trace of an algebraic number

Let $\alpha$ have minimal polynomial $m(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0.$ Show that \begin{equation*} \mathrm{Tr}(\alpha^k) = -k a_{n-k} - \sum_{i=1}^{k-1} a_{n-i} ...
0
votes
0answers
13 views

Adjoining a separable element to a field makes the extension separable

Let $k$ be a field, and $K$ be an extension. Suppose $a\in K$ is separable over $k$. What's a clever way to show that $k(a)$ is separable (i.e., that all elements of $k(a)$ have separable minimal ...
4
votes
1answer
48 views

Norm on “tower” of fields (the question comes from algebraic number theory)

Consider the field extensions $L\supset K'\supset K$, where both $L|K$ and $K'|K$ are finite and Galois. I want to prove that $$N_{L|K}(L^\ast)\subseteq N_{L|K'}(L^{\ast})$$ Maybe it is very easy, ...
1
vote
1answer
17 views

Find the fixed field of the following subgroup?

I am trying to understand some concepts via random exercises I found from past papers but this particular one, I am not sure even where to start. There aren't any solutions for the paper so would ...
0
votes
0answers
15 views

The relationship between “irreducible polynomials” “$K$-automorphisms” and “degree of extension”

I know what each of them are but I don't see how they strongly tie-in. I am looking at this here Let the field extension be $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}$. Then $t^2-5$ is ...
2
votes
1answer
32 views

I do not understand the definition of a “K-automorphism”

I am reading Ian Stewart's Galois Theory and frankly, the following definition is puzzling me Let $L:K$ be a field extension, so that $K$ is a subfield of the subfield $L$ of $\mathbb{C}$. A ...
1
vote
1answer
16 views

Finite Galois Field extension of a field $F$ containing all roots of unity

Let $F$ be a field that contains all roots of unity. Furthermore, let $K$ be a finite algebraic extension of $F$ with abelian Galois group . Then $$K= F(z_1,\ldots , z_n)$$ for some $z_i \in K$ ...
2
votes
2answers
24 views

Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. Find an element $Y \in K(X)$ so that $K(Y)=\{ f\in K(X) : \phi (f)=f\}$.

Let $K(X)$ be the field of rational functions of $X$ over some field $K$. Let $\phi: K(X) \rightarrow K(X)$ be the K-morphism such that $\phi (X)=1-X$. We have $L:=\{ f\in K(X) : \phi (f)=f\}$. Find ...
0
votes
1answer
30 views

Can the galois group be the symmetric group, if the discriminant is a perfect square?

Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square. Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of ...
7
votes
1answer
65 views

Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
2
votes
0answers
38 views

Fixed fields in Neukirch's book (chap. IV): notational problem

I am reading chapter IV of Neukirch's ANT, and there is a thing that I don't understand. First of all I have to introduce the notations of chapter IV. $G$ is a profinite group and: Clearly this ...
1
vote
1answer
30 views

Stewart's “Galois Theory”

One of the questions in Stewart's "Galois Theory" says: Let $p(t) \in \mathbb{Q}[t]$. Show that $p(t)$ has a unique expression of the form: $p(t)=(t-\alpha_1)\cdot \cdot \cdot (t- \alpha_r)q(t)$ ...
1
vote
1answer
34 views

Basic question in Galois theory (on applying elements of the Galois group to a root of polynomial)

Suppose I have $K = \mathbb{Q}(\theta)$ and let $f$ be the minimal polynomial of $f$ over $\mathbb{Q}$. Suppose $f$ has degree $n$ so that the degree of $K$ over $\mathbb{Q}$ is $n$. Suppose further ...
1
vote
1answer
22 views

$K(x)$ not stable relative to $K(x,y)$ and $K$

Prove that in the extension of an infinite field $K$ by $K(x,y)$, the intermediate field $K(x)$ is Galois over K, but not stable (relative to $K(x, y)$ and $K$). I know that if K(x) is algebraic it ...
1
vote
0answers
28 views

Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
0
votes
1answer
85 views

Finding Galois Group

Let $\mathbb Q\subset\mathbb Q(\sqrt{3}+\sqrt[3]{5})$. I want to find the Galois group of the given field extension. It would be easy for me if I could find a basis of the given field extension ...
1
vote
0answers
35 views

Splitting field of a polynomial has a solvable group of automorphisms

Prove that splitting field of a polynomial $f \in \mathbb{k}[x]$ ($f = 0$ solvable by radicals) has a solvable group of automorphisms $Aut_\mathbb{k}(\mathbb{F})$. I have just started learning Galois ...
1
vote
1answer
42 views

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$ over the field $\mathbb{k}_{sym}(x_1,\dots,x_n)$, the characteristic of $\mathbb{k}$ is 2 and $$D(x_1,\dots,x_n) = \prod_{1 \leq ...
2
votes
1answer
25 views

Symmetric Polynomial in roots is in $F[X]$

I recently came across the following claim. Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define $$ g_h = \prod_{1\le ...
2
votes
1answer
48 views

Galois group of $\mathbb Q(\sqrt{4+\sqrt 7})/\mathbb Q$

Let $\alpha =\sqrt{4+\sqrt 7}$ and $\beta =\sqrt{4-\sqrt 7 }$. I have to compute $Gal(\mathbb Q(\alpha )/\mathbb Q)$. So, I found that $Gal(\mathbb Q(\alpha )/\mathbb Q)=\mathbb Z/2\mathbb Z\times ...
2
votes
1answer
28 views

Prove that $f = q_1 + Gq_2$ for some $q_1, q_2 \in \mathbb{k}_{sym}(x_1,\dots,x_n)$

Let the orbit of the function $f \in \mathbb{k_(x_1,\dots,x_n)}$ under the action of $\{\phi_\sigma|\sigma \in \mathfrak{S}_n\}$ has length 2. Prove that $f = q_1 + Gq_2$ for some $q_1, q_2 \in ...
0
votes
1answer
41 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
2
votes
0answers
24 views

Find isomorphism between Kummer's field and filed of n-th roots

Let $\mathbb{k}$ be the field of all the $n$th roots of $1$ and $\mathbb{F=k}(\alpha_1, ..., \alpha_n)$ is a Kummer's field over $\mathbb{k}$, where $\alpha^n_i=a_i \in \mathbb{k^*}$, $i=1,\dots,m$. ...
3
votes
0answers
35 views

Finite separable normal extension has Galois abelian group

Prove that finite separable normal extension $\mathbb{F}$ of field $\mathbb{k}$, $\operatorname{char}(\mathbb{k})=p > 0$ has Galois abelian group $\operatorname{Gal}(\mathbb{F}/\mathbb{k})$ with ...
1
vote
1answer
32 views

Describe the group of $\operatorname{Aut}_{K(x^p, y^q)}(K(x, y))$

Let $K$ be a field, $\operatorname{char}K = p > 0$, $q$ is prime such that $p-1 \equiv 0 \pmod{q}$. Describe the group of $\operatorname{Aut}_{K(x^p, y^q)}(K(x, y))$. It's pretty clear that we ...
1
vote
1answer
22 views

Can I find a Galois extension which contains a finite set of algebraic elements?

Suppose $K/F$ is an algebraic extension of fields. Pick a finite collection $a_1,...,a_n \in K$. Can I find a Galois extension of $F$ which contains these $n$ elements of $K$?
3
votes
1answer
32 views

Outer Automorphisms of Galois groups

Assume $K/\mathbb{Q}$ is Galois extension of number fields and let $F \subset K$. Let $Gal(K/F) \cong G$ and let $\sigma$ be an outer automorphism of $G$ as an abstract group. Is there a way to ...
2
votes
1answer
41 views

Suppose $\phi\in{Aut(C)}$ and continuous, why $\phi$ must fix $R$? [closed]

Suppose $\phi\in{Aut(\mathbb{C})}$ and continuous, why $\phi$ must fix $\mathbb{R}$? I know that continuity is crucially important here, but I'm not sure how.
1
vote
2answers
62 views

Isn't the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}(\sqrt3))$ just the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2))$?

Isn't the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}(\sqrt3))$ just the structure of $\mathrm{Gal}(\mathbb{Q}(\sqrt2))$? I'm getting confused about common algebra notation. ...
-1
votes
1answer
36 views

How many orbits are possible in the group action?

Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then what is the number of orbits for action of $G$ on the field with nine elements?
0
votes
1answer
39 views

Multiplicative Inverse in a $256$ Galois Field

I am working on finding the multiplicative reverse in $GF(2^8)$ using the Euclidean Algorithm but after reading multiple sources, I feel as though I am proceeding incorrectly. Using the irreducible ...
0
votes
2answers
48 views

Minimal Galois extension, describe structure of $Gal(L/\mathbb Q)$

Find the minimal Galois extension $L$ of $\mathbb Q$ containing $\mathbb Q(\sqrt[4]{5})$. Describe the structure of $Gal(L/\mathbb Q)$. I think $L$ is a splitting field of $X^4-5$ over $\mathbb Q$. ...
3
votes
1answer
52 views

Is the tensor product of 2 finite extension of $\Bbb Q$ isomorphic to a direct sum of fields?

I have $K_1$ and $K_2$ two finite extensions of $\Bbb Q$. I can construct $K_1 \otimes_\Bbb Q K_2$. This is clearly isomorphic to a direct sum of field as vector space (indeed one can easily see that ...
2
votes
2answers
49 views

Why are the elements of a galois/finite field represented as polynomials?

I'm new to finite fields - I have been watching various lectures and reading about them, but I'm missing a step. I can understand what a group, ring field and prime field is, no problem. But when we ...
0
votes
1answer
20 views

Trisecting angle equivalence of constructing a segment

After reading Wikipedia and some previous questions asked in this site, I still don't understand this. Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). ...
1
vote
1answer
25 views

Completion of a Galois Extension $K | \mathbb{Q}$ w.r.t. to a non archimedean abs. value is a Galois extension

As the title suggests, I'm interested in a proof of the following fact: Let $K | \mathbb{Q}$ be a finite Galois extension, and let $\mathfrak{p}$ be a prime ideal of the ring of integers of $K$. ...
1
vote
2answers
32 views

Finite extensions of $\mathbb F_p(t)$ [on hold]

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$. Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some ...
2
votes
0answers
69 views

How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt ...
4
votes
1answer
32 views

Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where ...
0
votes
0answers
30 views

Minimal cyclotomic field containing a given quadratic field?

There was an exercise labeled difficile (English: difficult) in the material without solution: Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural ...
0
votes
0answers
11 views

$V\otimes_kK\cong \oplus We_i$?

Let $K/k$ be a Galois extension with Galois group $G$, let $V$ be a $K$-vector space with semi-linear $G$ action, which means: $G\to Aut(V_k)$, $g\mapsto r_g$ such that $r_1=Id, r_{gh}=r_g\circ r_h$, ...
1
vote
1answer
36 views

Finding Galois conjugates

I'm working on a big exercise from Dummit & Foote (p.584) with the end goal of constructing a polynomial with Galois group $Q_8$ (Quaternion group of order $8$). Take $$\alpha = ...
1
vote
2answers
63 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
1
vote
1answer
34 views

Basic questions about field homomorphism extension

I learned that one can extend the homomorphism "injection" $k\hookrightarrow \Omega$ (algebraic closure) to a morphism $u:k[a]\to \Omega$ where $a\in \Omega$ is algebraic over $k$ such that the ...
0
votes
2answers
38 views

Show that sum of roots is rational.

If $f(X)$ in $\mathbb{Q}[X]$ is an irreducible polynomial polynomial of degree $n \geq 2,$ with roots $\alpha_1, \alpha_2,\ldots,\alpha_n$ in $\mathbb{C},$ show that $\displaystyle ...
1
vote
1answer
31 views

Galois group of a polynomial over $\mathbb{C}[t]$

To find the Galois group of the polynomial $X^3-X-t\in\mathbb{C}[t]$, an approach is to compute the discriminant (equal $(2-\sqrt{27}t)(2+\sqrt{27}t)$) which is not a square in $\mathbb{C}[t]$ so the ...
0
votes
1answer
28 views

Invariant subfields and Galois group

Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Let $\alpha\in K$ be a root of $f$. Let $H$ be the subgroup of $G$ that ...
1
vote
1answer
45 views

Intersection of Kummer extension [closed]

Let $p$ and $q$ be two prime numbers and $\omega$ be the primitive 3rd root of unity. The splitting field of $X^3-p$ over $\mathbb{Q}$ is $K_p=\mathbb{Q}(p^{\frac{1}{3}},\omega)$ and we have a similar ...
4
votes
3answers
52 views

Monic irreducible polynomials of degree 6 in $F_{5}[X]$

Question A How many monic irreducible polynomials of degree 6 in $F_{5}[X]$ Question B Give an example of an irreducible polynomial of degree 6 in $F_{5}[X]$ Idea for a Such a polynomial would be ...
2
votes
2answers
90 views

General technique for finding minimal polynomial? [closed]

I always have a lot of trouble with these problems, "find the minimal polynomial of {number} over {field}" What are the general procedures for solving problems of this format? Thank you for your ...