1
vote
1answer
50 views

What is $\hat{\mathbb{Z}}$?

I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes ...
1
vote
0answers
22 views

determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
4
votes
2answers
37 views

Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
1
vote
0answers
16 views

Finite field extensions as projective group algebras

For a finite field extension $K \subset L$, can $L$ always be seen as a projective group algebra over $K$? That is, does there always exist a finite group $G$ and isomorphism $L \simeq K _\alpha [G]$, ...
2
votes
1answer
41 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
1
vote
3answers
44 views

Extensions of degree $1$.

My doubt is very simple: Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$? I'm trying to prove the equality without success. Thanks in advance
4
votes
1answer
65 views

Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
0
votes
2answers
29 views

Primitive element of the fixed field of a subgroup of the galois group of a prime cyclotomic extension

This is a question on one step in a proof from Dummit and Foote pg. 597. Let $p$ be an odd prime (there isn't much to say for the $p=2$ case) and let $G=\text{Gal}(\mathbb{Q(\zeta_p)}/\mathbb{Q})$ ...
1
vote
1answer
26 views

Prove that $K=k(\alpha)$

Prove: If K\k is a Galois extension and $\alpha \in K$ with $\sigma(\alpha)\neq \alpha$ for all $\sigma \in Gal(K,k)\backslash \lbrace id_K \rbrace$, than $K=k(\alpha)$.
1
vote
0answers
30 views

Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
1
vote
1answer
49 views

Show that a sequence of fields exists

I do not have a clue how to solve the following problem: Let $K\subseteq L$ be Galois extension of degree $p^n$, where $p$ is prime and $n$ is natural. Show that there exists a sequence of subfields ...
1
vote
0answers
33 views

Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...
6
votes
2answers
63 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
0
votes
0answers
22 views

Show that $K_1K_2=K_1(K_2)$ is abelian. [duplicate]

Let $L/F$ be a field extension and $L/K_i/F$ with $K_i/F$ abelian. Show that $K_1K_2=K_1(K_2)$ is abelian.
2
votes
2answers
72 views

Continuation on: Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension?

Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension? People seem pretty convinced this is Galois. I present an issue I am having though. So the monic polynomial p(x) in ...
2
votes
2answers
35 views

Compute the Galois group for the root field of the polynomial $x^3$+$2x$+2 over $Z_3$

Compute the Galois group for the root field of the polynomial $x^3$+$2x$+2 over $Z_3$ Choose a root $\alpha$ in the root field of the polynomial. We found three roots in this root field, $\alpha$, ...
1
vote
0answers
23 views

A field without the extension property

A field $\mathbb{K}$ is said to have the extension property if every automorphism of $\mathbb{K}(t)$ is an extension of an automorphism of $\mathbb{K}$, where $t$ is a variable. It is equivalent to ...
0
votes
1answer
96 views

Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension?

Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension? Text: Essentials of Modern Algebra, Cheryl Chute Miller
0
votes
2answers
36 views

Given $u=e^{i2\pi\over{7}}$, determine whether $\mathbb{Q}(u)/\mathbb{Q}$ is a Galois extension

I'm a bit confused because I tried this: $u=e^{i2\pi\over{7}}=(e^{i2\pi})^{1\over7}=(\cos2\pi+i\sin2\pi)^{1\over7}=1$
2
votes
1answer
86 views

Find the Galois group

Could you help me in finding the Galois group of $\mathbb{Q}\left(\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3}}\right)$ over $\mathbb{Q}$? I can only say that $\mathbb{Q}(\sqrt{2}) /\mathbb{Q}$ ...
3
votes
2answers
41 views

Extending Automorphisms, Isomorphisms

Suppose we have an isomorphism $\varphi: K_1 \to K_2$ where there is a base field $F$ and a Galois extension field $L$, ie, $F \subsetneq K_1, K_2 \subsetneq L$, $\varphi|_F = Id$ and $|Gal(L/F)| ...
4
votes
1answer
60 views

Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial ...
4
votes
1answer
47 views

A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
2
votes
1answer
23 views

Proving that $\mid Gal(E/F)\mid\leq [E:F]$ by induction

I am looking for help understanding the proof that $\mid Gal(E/F)\mid\leq [E:F]$ by induction on the degree of the extension. $E/F$ is a finite algebraic extension. I'll mark them by [1], [2], [3]. I ...
4
votes
2answers
106 views

Galois group - extend homomorphism to automorphism

Let $K \subset L$ be a finite Galois extension, $M$ a field with $K \subset M \subset L$ and $G := \text{Aut}(L/K)$. I want to show that if $\sigma \, \colon M \longrightarrow L$ is a ...
18
votes
3answers
326 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
2
votes
1answer
79 views

$L=K(\alpha)$ with $\alpha^p-\alpha=a\in K$

Let $K$ be a field of characteristic $p$. $L=K(\alpha)$ with $\alpha^p-\alpha=a\in K$, an extension of order $p$. Show that there does not exist $\beta \in L$ such that $\alpha^{p-1}=\beta ^p-\beta$. ...
0
votes
0answers
25 views

E is a Galois extension of F

15.32 Let $E$ be a finite extension of a field $F$ with dimension $n$. Show that $|\operatorname{Gal}_F(E)| = n$ if, and only if, $E$ is a Galois extension of $F$. Please some help to find the ...
3
votes
1answer
50 views

A question about Galois extension

Let $n\geq 2$. Let $K$ be a field which contains $n$ distinct n-th roots of unity. Let $L=K(\sqrt[n]{a})$ with $a\in K$ and $[L:K]=n$. Show: there exists a Galois extension $K\hookrightarrow M$ such ...
3
votes
0answers
131 views

If $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable.

In all the books that I've checked the use of roots of unity (as hypothesis) is very crucial to prove that if $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable, but ...
0
votes
1answer
14 views

There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...
1
vote
1answer
30 views

maximal abelian extension of exponent $q-1$ of $\mathbb F_q((t))$

I would like to find the maximal abelian extension of exponent $q-1$ of $K=\mathbb F_q((t))$ and find its Galois group. Due to Kummer theory this extension is $K(\sqrt[q-1]{K^*})$ and it's Galois ...
0
votes
1answer
21 views

irreducible polynomial is still irreducible on purely inseparable extension

I wish to prove that $\alpha \in C$ where $C$ is algebraically closed field of $F$, $f(t) \in F[t]$ is minimal monic irreducible, separable polynomial having $\alpha$ as a root. Then let ...
0
votes
1answer
61 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
3
votes
3answers
139 views

Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$?

Problem: Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $f(x) = x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$? My Attempt: I have proved that ...
0
votes
1answer
39 views

Prove every finite extension of subfields of $\mathbb{C}$ is simple

This is a question is from a past paper, and it's only worth 8 marks, and it's really got me. I'm allowed to assume that if $L/K$ is such an extension, that there are $[L:K]$ $K$-embeddings of $L$ ...
1
vote
1answer
90 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
2answers
41 views

Prove that every finite group occurs as the Galois group of a field extension of the form $F(x_{1}, \dots , x_{n})/F$.

I've seen that every abelian group is a Galois group over $\mathbb{Q}$ for some subfield of a cyclotomic field, but I'm not sure about this more general result.
5
votes
2answers
81 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 ...
2
votes
1answer
67 views

If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?

Defn: A field extension $K/F$ is obtained by adjoining $n$th roots if there is a tower of fields $F=K_1\subset\cdots\subset K_n=K$ such that for each $i$, $K_{i+1}=K_i(\alpha_i)$ and there exists ...
0
votes
1answer
65 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 ≠ ...
0
votes
2answers
65 views

Construct an embedding

I'm dealing with this problem from the book "Field Theory" (Steven Roman) Suppose $F$ and $E$ are fields and $\sigma : F \rightarrow E $ is an embedding. Construct an extension of $F$ that is ...
3
votes
1answer
106 views

$\mathbb{Q}(\sqrt[p]{q}) \neq \mathbb{Q}(\sqrt[p]{r})$ for $p,q,r$ primes and $q \neq r$.

Let $p,q$ and $r$ be primes in $\mathbb{Z}$ with $q \neq r$. Let $\sqrt[p]{q}$ denote any root of $x^p-q$ and let $\sqrt[p]{r}$ denote any root of $x^p - r$. I need to prove that ...
6
votes
3answers
95 views

Showing that $\mathbb{Q}(\zeta_p, \sqrt[p]{\ell}) = \mathbb{Q}(\zeta_p + \sqrt[p]{\ell})$ for $p,\ell$ primes.

We consider the polynomial $x^p - \ell$, where $p,\ell$ are both prime numbers. Let $\zeta_p$ be a $p$-th root of unity. We wish to show that $L = \mathbb{Q}(\zeta_p, \sqrt[p]{\ell})$ is the same as ...
0
votes
0answers
58 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 ...
0
votes
1answer
41 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 ...
3
votes
0answers
61 views

Does cyclic field imply Galois?

I am thinking about the following statement, and I wonder if this is true: Every cyclic field is Galois. (we are in characteristics $0$). I have started with a cubic case and tried to make use of the ...
6
votes
1answer
262 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not 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
59 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 ...
2
votes
1answer
108 views

Field not closed under complex conjugation

What is an example of an algebraic field not closed under complex conjugation? In all subfields of $\mathbb C$ I think of, complex conjugation is a transposition. I think I understand that it is ...