1
vote
1answer
36 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
28 views

So-called Artin-Schreier Extension

Let $F$ be a field of characteristic $p$. Let $K$ be a cyclic extension of $F$ of degree $p$. Prove that $K=F(\alpha)$ where $\alpha$ is a root of the polynomial $p(x) = x^{p} - x - a$ for $a \in ...
1
vote
1answer
42 views

Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
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 ...
1
vote
0answers
39 views

Determine the fixed field of a subgroup of $S_3$ and check the Galois correspondence

I came across this homework problem and I'm stumped. This is the third part of the problem, so to give some context, here's what we have so far: Let $f(x)=x^3-2 \in \mathbb{Q}[x]$. We know that the ...
3
votes
2answers
53 views

Prove that the Galois group of $x^n-1$ is abelian over the rationals

If $p(x)=x^n-1$, prove that the Galois group of $p(x)$ over the field of rational numbers is abelian. Here's what I have so far. Denote the Galois group $G(K,\mathbb{Q})$, where $K$ is the ...
1
vote
2answers
106 views

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
1
vote
1answer
65 views

The automorphism group of a fixed field

Let $E$ be a Galois extension of $F$ with Galois group $G$, and let $L$ be the fixed field of a subgroup $H$ of $G$. Show that the automorphism group of $L/F$ is $N/H$ where $N$ is the normalizer of ...
1
vote
1answer
44 views

No field extension is “degree 4 away from an algebraic closure”

I have seen this problem asked by another user but it isn't completely solved in the answers. I'm trying to do it, but I can't. Question: Suppose $[L:K]=4$ and $charK≠2$ and $L$ is algebraically ...
5
votes
1answer
90 views

Homework: No field extension is “degree 4 away from an algebraic closure”

Question: Suppose $[L:K]=4$ and char$K \neq 2$ and $L$ is algebraically closed. Show that there is an intermediate field $M$ such that $[L:M]=2$ and that $X^2 + 1$ splits over $M$. Show that this ...
2
votes
1answer
35 views

Degree of field extension $[F(\alpha_1+\alpha_2):F]$

Given irreducible quartic $f(x) \in F[x]$ with roots $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and Galois group $G = S_4$, what is the degree of the extension $E = F(\alpha_1+\alpha_2)$ over $F$? Find ...
1
vote
2answers
106 views

Find the value of $\cos(2\pi /5)$ using radicals [duplicate]

This is homework so if there is another example that can illustrate the technique I would happily accept that as guidance. The only thing I have been able to find is a question asking about ...
1
vote
0answers
69 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
71 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
46 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
37 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
29 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.
0
votes
1answer
45 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
90 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. ...
0
votes
0answers
80 views

Show that $F/K$ is algebraic implies $F(x)/L(x)$ is algebraic (extension)

(1.) Show that if F is an algebraic extension of $K$, then $F(x)$ is an algebraic extension of $K(x)$. (2.) Show $[K:L] = [K(x):L(x)]$ (Attempt at solution): Let f/g be a rational function in ...
1
vote
2answers
84 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
48 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 ...
0
votes
1answer
68 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
23 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) ...
7
votes
1answer
132 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
87 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 ...
2
votes
0answers
63 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
0answers
42 views

First cohomology of a Galois group with finite base field

Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology. Then ...
2
votes
2answers
80 views

Finding a polynomial with given Galois group

In Galois theory we have just been introduced to the fundemental theorem. We were then assigned this homework problem as a sort of inverse problem to the typical "Find the Galois group of $f$ ": ...
4
votes
3answers
116 views

Is $\mathbb{Q}(5^{1/3})$ a Galois extension over $\mathbb{Q}$?

I am trying to prove or disprove that the simple extension $\mathbb{Q}(5^{1/3})$ is Galois over $\mathbb{Q}$. I suspect that this extension is not Galois, because an extension if Galois over ...
2
votes
1answer
67 views

Cyclotomic field automorphisms “fill up” $\mathbb{Z}/n\mathbb{Z}$?

I know from reading that the Galois group of a cyclotomic polynomial is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$. While I believe this, I can't figure out why that should work. In particular, it's ...
1
vote
1answer
70 views

What's wrong here when I compute $\operatorname{Gal} (x^4 -2 / \mathbb{Q})$

Maybe that's a stupid question and I'm missing something very trivial. Let $f(x) = x^4 - 2$ and $\alpha_1 = \sqrt[4]2, \alpha_2 = -\sqrt[4]2, \alpha_3 = i\sqrt[4]2, \alpha_4 = -i\sqrt[4]2$ be its ...
1
vote
1answer
55 views

Galois group of a reducible polynomial over a arbitrary field

How to proceed to determine the Galois group of a reducible polynomial over a field $F$. As an example I tried to compute the Galois group of $f(X)=X^4+4\in\mathbb{Q}[X]$; one can check that ...
4
votes
1answer
171 views

splitting field of $(x^3-2)(x^3-3)$ over $\mathbb Q$

Question: What is the Galois group of $f(x)=(x^3-2)(x^3-3)$ over $\mathbb Q$, and what are the subfields which contain $\mathbb Q(\zeta_3)$? The roots of $f(x)$ are ...
0
votes
3answers
115 views

Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$

Find the Galois group $\operatorname{Gal}(f/\mathbb{Q})$ of the polynomial $f(x)=(x^2+3)(x^2-2)$. Any explanations during the demonstration, will be appreciated. Thanks!
2
votes
0answers
91 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
1
vote
0answers
75 views

Show that an field extension is algebraic (normal).

Let $A/K$ be a field extension, I wanted to proof: $A/K$ is normal iff for every irreducible polynomial $P \in K[x]$ which has a root in $A$, the field extension $A$ contains a splitting field for ...
5
votes
2answers
115 views

existence of an automorphism of $k^a$ whose fixed field is $k$

Let $k$ be a field such that every finite extension is cyclic. Show that there is an automorphism of $k^a$ over $k$ whose fixed field is $k$. Here $k^a$ is the algebraic closure of $k$. P.S. It's a ...
8
votes
1answer
221 views

Galois group of $X^5 - X^3 - 2X^2 - 2X - 1$ over $\mathbb{Q}$.

So far I have found that this polynomial is irreducible, and its discriminant is a square. Therefore the Galois group it a transitive subgroup of $A_5$. I found out that the only transitive ...
2
votes
0answers
38 views

Prove (or disprove) that the Galois group of $X^4 - X^3 - 7X + 19$ is not $S_4$.

I have checked that $X^4 - X^3 - 7X + 19$ is coprime to its derivative. So it is separable. So I know that the Galois group of $X^4 - X^3 - 7X + 19$ must be a subgroup of $S_4$. But I need to prove ...
1
vote
1answer
42 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
4
votes
1answer
46 views

No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$

Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
3
votes
1answer
186 views

Proving $ x^p - a $ is irreducible iff $ a $ is not a $p$-th power of an element

I have the following homework question, which I've managed to do the forward implication of, but not the other direction: Let $\mathbb{K}$ be a field, $ a\in\mathbb{K}$ and $p$ be a prime. Show that ...
5
votes
2answers
221 views

Radical extension

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 $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
5
votes
0answers
47 views

Galois automorphisms and Field extensions [duplicate]

Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. I want to show that $\alpha^2+\beta$ has degree $9$. There are many ways to do this, but I wish to solve the problem ...
5
votes
1answer
234 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
1
vote
2answers
155 views

$M /K \land L /K$ algebraic $\implies ML/K$ algebraic?

Let $K \subset M$, $L\subset K'$, and let $ML$ denote the subfield of $K'$ generated by $M$ and $L$. Is the following true? $M/K$ and $L/K$ algebraic $\implies ML/K$ algebraic? Any hints proof or ...
12
votes
4answers
271 views

Galois Theory and Galois Groups

Show that $\mathbb{Q}[x]/\langle x^{3}-2\rangle = [{a + b\alpha + c\alpha^{2}: a, b, c \in \mathbb{Q}, \alpha^{3} = 2}]$ is not a Galois extension of $\mathbb{Q}$. In particular, show that every ...
2
votes
2answers
273 views

Galois Group of $x^5+1$

I need help to find the Galois Group of $x^5 +1$. I know that it has a 5-cycle and a 4 cycle and is not a subgroup of $A_5$. Thanks!
16
votes
3answers
266 views

How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?

Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...