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Questions tagged [galois-extensions]

For questions about Galois extensions of fields. We say that an algebraic extension $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

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A problem about the fixed field of a Galois extension

Let $L/K$ be a finite Galois extension of fields and $B\supseteq A$ are both Dedekind rings whose fraction fields are $L, K$ respectively. Given $\mathfrak{p}$ a prime ideal in $A$, define $T_{\...
Bowei Tang's user avatar
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1 vote
1 answer
65 views

Inertia field example of $ \mathbb{Q}_5(\sqrt[4]{50})$

Let $L = \mathbb{Q}_5(\sqrt[4]{50})$ and denote by $E$ the inertia field of the extension $L / \mathbb{Q}_5$. Write down a prime element $\pi_E $ of $ \mathcal{O}_E $ with $L = E(\sqrt{\pi_E})$. Can ...
Christian Schwacke's user avatar
3 votes
0 answers
92 views

Why $\mathbb{Q}(i,\sin(\frac{2\pi}{n}))=\mathbb{Q}(i,\zeta_n)$ if $n$ is odd?

I was trying to prove some properties to obtain that the degree of the extension of field $\mathbb{Q}(\sin(\frac{2\pi}{n}))\vert\mathbb{Q}$ is $\varphi(n)$ when $n$ is an odd positive integer (being $\...
Pablo Garcia Pastor's user avatar
1 vote
1 answer
79 views

Galois group of $Q(e^{2\pi i/14})/Q$

I want to know the Galois group of $Q(e^{2\pi i/14})/Q$. More precisely, I want to know how one can find the order of the Galois group and the automorphisms. First, I know that $x^{14}-1=(x^7 -1)(x^7 +...
Andrei's user avatar
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3 votes
2 answers
350 views

A question about Hilbert Theorem 90 and Artin-Schreier Theorem

I'm reading Lang's "Algebra" and there's a passage in the proof of Theorem 6.3 pg.290 (namely Hilbert's Theorem 90 additive form) for which I can't find a justification, if anyone could ...
F. Salviati's user avatar
0 votes
1 answer
74 views

Galois group of $X^3-X+1$ over $\mathbb{Q}$ and $\mathbb{R}$ without discriminant.

Yesterday I had an exam and I had to find the galois group of the polynomial $f = x^3-x+1$. My answer was $A_3$ which is probably wrong. First of all it has no roots by the rational root theorem so it ...
muhammed gunes's user avatar
4 votes
2 answers
118 views

Alternative method : $\mathbb{Q}(\zeta_3+\sqrt[3]{7})=\mathbb{Q}(\zeta_3,\sqrt[3]{7})$

Let $\zeta=\zeta_3$ be a third root of unity. I want to proof that $\mathbb{Q}(\zeta+\sqrt[3]{7})=\mathbb{Q}(\zeta,\sqrt[3]{7})$. One inclusion is clear that is : $\mathbb{Q}(\zeta+\sqrt[3]{7})\subset\...
muhammed gunes's user avatar
5 votes
4 answers
114 views

Any direct method to show that $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{5}]$?

We know that Galois extension is simple extension, so the splitting field of $(X^2-2)(X^2-3)(X^2-5)$ over $\mathbb{Q}$ satisfies $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\alpha]$, for some ...
shwsq's user avatar
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0 answers
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How to find the Galois group of an infinite Galois extension? For example $\mathbb{Q}(\mathbb{\sqrt{\mathbb{Q}}})$. [duplicate]

I know that this is a very broad question, but I am especially interest in the case of the Galois extension $\mathbb{Q}(\sqrt{\mathbb{Q}}) = \mathbb{Q}( \{ \sqrt{-1} \} \cup \sqrt{p} \mid p \text{ ...
Cosima's user avatar
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0 votes
0 answers
153 views

Help me to verify the proof of this theorem, which is proving $k$ª $=$ $\prod_{i=1} ^{\infty}$ $\mathbf K_{p_i}$ by using maximality and minimality

I want to verify my proof is true or false. The exercise what I want to prove is under theorem. $\mathbf {Exercise}$: Let $\mathbf k$ be some perfect field and $\mathbf K_{p_i}$ is a compositum of ...
Snailman's user avatar
3 votes
1 answer
74 views

Constructing a primitive element for each fixed field of a Galois extension

Let $K/F$ be a finite Galois extension with Galois group $G$ and normal basis $ \{\sigma(\alpha):\sigma\in G\} $ and let $H\leq G$ be a subgroup. I am asked to show that that $F(Tr_{K/K^H}(\alpha))=K^...
A Name's user avatar
  • 316
3 votes
1 answer
105 views

non-isomorphic number fields

Show that following 2 polynomial do not generate isomorphic number number fields: $P_1(x)= x^3+10x+1$ $P_2(x) = x^3-8x+15$ I see $Disc(P_1)=Disc(P_2)=-4027$ and both of them have a real root and 2 ...
mshj's user avatar
  • 520
0 votes
0 answers
38 views

Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$

So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
NoetherBoy 's user avatar
1 vote
1 answer
46 views

Does the Galois group of a polynomial change upon translation?

I know that a polynomial $f(X)$ is irreducible iff $f(X+1)$ is irreducible. Is it true for the Galois group? I think it should but I don't know if there ir a neat proof (I have thought of writing the ...
Valere's user avatar
  • 1,344
0 votes
0 answers
59 views

Lemma 13, Section 5.2 of Hungerford’s Algebra

Theorem 2.2. Let $F$ be an extension field of $K$ and $f\in K[x]$. If $u\in F$ is a root of $f$ and $\sigma \in \text{Aut}_K F$, then $\sigma (u)\in F$ is also a root of $f$. If $F$ is an extension ...
user264745's user avatar
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0 votes
1 answer
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What is the fixed field of $\mathbb{Q}(\sqrt[3]{5}, \sqrt[3]{5}\zeta_3)$ with 3-cycle group?

While doing an exercise I am asked to calculate the fixed field of $$L:=\mathbb{Q}(\sqrt[3]{5}, \sqrt[3]{5}\zeta_3)$$ by the subgroup $$\langle (a,b,c) \rangle := \langle (\sqrt[3]{5}, \sqrt[3]{5}\...
Flynn Fehre's user avatar
2 votes
2 answers
62 views

$K_1,K_2$ are isomorphic subfields of $L$ via $\sigma$, every polynomial $P(x)$ over $K_1$ have the same number of roots in $L$ as $\sigma(P(x))$?

Let $K_1,K_2$ be isomorphic subfields of $L$ via $\sigma:K_1\to K_2$, must every irreducible polynomial $P(x)$ over $K_1$ have the same number of roots in $L$ as $\sigma(P(x))$?
Z Wu's user avatar
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0 answers
37 views

Do we always have $|\mathrm{Mor}_K(K(\alpha),F)|$ divide $|\mathrm{Mor}_K(K(\alpha),\overline{K})|$ with $\alpha\in F$ algebraic over $K$?

Let $F/K$ be an algebraic field extension and $\alpha\in F$. Let $m(x)$ be the minimal polynomial of $\alpha$ over $K$. Then $|\mathrm{Mor}_K(K(\alpha),F)|$ is the size of the roots of $m(x)$ in $F$ ...
Z Wu's user avatar
  • 1,785
2 votes
1 answer
45 views

KL is an abelian extension implies K and L are abelian extensions

Prove of give a counterexample: If $KL$ is abelian then $K$ and $L$ are abelian. I have proven the converse, that is if $K$ and $L$ are abelian then $KL$ is abelian. I have a feeling the above doens'...
JLGL's user avatar
  • 795
1 vote
1 answer
39 views

Primitive element of Galois extension whose conjugates are linear dependent.

I was reading in my fields and Galois theory course about normal basis for Galois extensions and primitive elements. The section about normal basis opened the chapter saying that we can have a ...
IAG's user avatar
  • 223
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0 answers
31 views

Is there anywhere a list of the groups $Gal(\mathbb{Q}(\xi_n)/\mathbb{Q})$? [duplicate]

As a math undegraduate, I was doing my homework of fields and Galois theory course and observed that many questions asked us to calculate the group $Gal(\mathbb{Q}(\xi_n)/\mathbb{Q})$, where $\xi_n$ ...
IAG's user avatar
  • 223
1 vote
0 answers
32 views

Inverse Galois Theory - Normal polynomials with given Galois group G but with in-equivalent actions on their roots

I am looking for an example of 2 irreducible normal polynomials $f(x), g(x) \in Q[x]$ which have the same Galois group $G$ over $Q$, but for which the action of $G$ on the roots of $f(x)$ and $g(x)$ ...
nor's user avatar
  • 121
0 votes
1 answer
26 views

Is the class of separable extensions distinguished?

We know, thanks to embeddings, that the class of separable extensions verified the property of the fields tower. That is, in a fields tower $K\subseteq F\subseteq E$, it is true that: $E/K$ is ...
IAG's user avatar
  • 223
0 votes
2 answers
51 views

Is the extension mod irreducible polynomial Galois?

Given a field $\mathbb{K}$ and an irreducible polynomial $p(x) \in \mathbb{K}[x]$, one can construct always a field in which $p(x)$ has a root, in particular $\mathbb{K}[x]/(p(x))$ (the quotient field)...
n-0's user avatar
  • 133
0 votes
1 answer
29 views

Confused about Proof of Theorem regarding equivalent conditions for a finite extension E/F with Galois Group G.

A Theorem in my textbook says the following: The following conditions are equivalent for a finite extension $E/F$. with Galois group $G=Gal(E/F)$: (i) $F=E^G$ (where $E^G$ are the elements of $E$ ...
Sachin's user avatar
  • 81
2 votes
1 answer
55 views

How to calculate Gal$(F(\mu_{p^\infty})/F(\mu_p))$ for a number field $F$?

Let $F$ be a number field. Recall that we define $$F(\mu_{p^\infty})=\bigcup_{n=1}^{\infty}F(\mu_{p^n}).$$ I want to calculate the group Gal$(F(\mu_{p^\infty})/F(\mu_p))$. I know that this is supposed ...
Rocket_Rabbit77's user avatar
2 votes
1 answer
60 views

Primitive element of a finite field whose powers do not lie inside the prime subfield

Let $p$ be a prime, and consider the finite field $\mathbb{F}_p$. Fix any $n\ge1$, and consider the field extension $\mathbb{F}_{p^n}/\mathbb{F}_p$. If $\alpha\in\mathbb{F}_{p^n}$ is a ...
The Discrete Guy's user avatar
0 votes
0 answers
19 views

problem with local Artin map in Hazewinkel article

$\def\cU{{\cal U}} \let\lra\longrightarrow$ I'm studying the Hazewinkel article "Local class field is easy" (a bit presomptuous as far as i'm concerned !). Readable here https://www....
noradan's user avatar
  • 309
1 vote
1 answer
64 views

Is it true that $i \in \mathbb{Q}(\sqrt{2},\xi)$, where $\xi = 1_{\frac{2\pi}{3}} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}$?

I am trying to do the following exercise: Give the Galois group of $\mathbb{Q}(\sqrt{3},\xi)$ over $\mathbb{Q}$, where $\xi = 1_{\frac{2\pi}{3}}$. Prove that $i \in \mathbb{Q}(\sqrt{2},\xi)$ and group ...
David's user avatar
  • 165
2 votes
1 answer
64 views

Some questions about the trace of $\alpha \in K$ from $K$ to $F$ where $K,F$ are fields

I am just beginning my study in field theory, and I am doing a question about the trace of $\alpha \in K$ from $K$ to $F$ where $K,F$ are fields: Let $K/F$ be a finite field extension and $\alpha ∈ K$...
ZYX's user avatar
  • 1,131
0 votes
1 answer
66 views

A question about the embedding from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to an algebraic closure of $\mathbb{Q}$

I am now just beginning my study in field theory and I am trying to find all embeddings from $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to $\bar{\mathbb{Q}}$ (an algebraic closure of $\mathbb{Q}$). Here, an ...
ZYX's user avatar
  • 1,131
0 votes
0 answers
25 views

Representatives of Quotient Group of Galois Groups

Let $M:L:K$ be normal and seperable field extensions, then we know by the fundamental theorem of Galois theory that $$ \frac{\text{Gal}(M:K)}{\text{Gal}(M:L)} \cong \text{Gal}(L:K) $$ I was wondering ...
PhPanda's user avatar
  • 315
0 votes
0 answers
30 views

Proof in The Galois Groups and Fundamental groups by Szamuely (lemma 3.4.2 )

I am currently reading Chapter 3.4 of Galois group and Fundamental group by of Szamuely and I am stuck at an important step of lemma 3.4.2. I am not sure I understand what they mean by a "...
fourofour_un's user avatar
1 vote
1 answer
30 views

galois group of completions

Let "ur" be for "unramified maximal". I read as "well known" that if $L/K$ is an abelian totaly ramified extension of the local field $K$ then $Gal(\widehat{L_{ur}}/\...
noradan's user avatar
  • 309
1 vote
1 answer
68 views

Construction of Type II optimal normal basis of $GF(2^n)$ over $GF(2)$

Substituting value $n=2$ in Construction theorem of Type II Optimal normal basis For $n=2$, ONB of Type II exist for $\mathbb{F}_{2^n}$ over $\mathbb{F}_2$. because, 2 is primitive $\mathbb{Z}_5$. By ...
Akhilesh Ajithan's user avatar
2 votes
1 answer
108 views

How to compute the different ideal of the cyclotomic field extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$? [closed]

Let $p$ be a prime number, $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field extension of $\mathbb{Q}$ by adding a $p$-th root of unity. There is a notation called different ideal, which is defined to ...
ZZP's user avatar
  • 150
0 votes
0 answers
25 views

Trascendental extension over the complex field

Let t be a variable and consider the extension over $\mathbb{C}$, $\mathbb{C}(t)/\mathbb{C}(t^{12})$. I have already proven that it's a Galois extension. I want to find the Galois group. I know it ...
Mikel's user avatar
  • 51
0 votes
1 answer
40 views

density of $\langle \textit{Frobenius}\rangle$ in $Gal(L/K)$

$K$ is a local field. $L/K$ is Galois with $K_{ur}\subset L$, where $K_{ur}$ is the maximal unramified extension of $K$. Noting $\Phi\in Gal(L/K)$ a lift of the Frobenius $\phi\in Gal(K_{ur}/K)$ and $...
noradan's user avatar
  • 309
1 vote
0 answers
52 views

What is known of this problem similar to the inverse Galois problem? Galois groups of $\mathbb{Q}(X)/\mathbb{Q}(h(X))$ for $h \in \mathbb{Q}(X)$

What is known about the answer to the following question? What finite groups can be realised as the Galois group of finite Galois extensions of the form: $$\mathbb{Q}(X)/\mathbb{Q}(h(X))$$ for $h(X)$ ...
Robin's user avatar
  • 3,940
2 votes
1 answer
125 views

$f\in\mathbb{Q}[x]$ is irreducible in $\mathbb{Q}[x]$ and the Galois group of $f$ has order 99 , what is the degree of $f$.

I need help for the following problem(what is the key-idea that problem): Problem: Let $f(x)$ be a monic polynomial with rational coefficients. Assume $f(x)$ is irreducible in $\mathbb{Q}[x]$ and the ...
TrItOs's user avatar
  • 111
1 vote
0 answers
65 views

Construct primitive element for intermediate field of cyclotomic field extension

I am studying subfields of cyclotomic extension, especially the following situation: Let $\xi$ be a primitive $m$-th root of unity, and $\chi$ be a multiplicative character on $(\mathbb{Z}/m\mathbb{Z})...
isz's user avatar
  • 31
0 votes
1 answer
145 views

If $E/F$ is a Galois extension, $Gal(E/F)$ is finite, and $K$ is an intermediate field that is a degree 2 extension from $F$, prove $K/F$ is Galois

I am a bit stuck on how I am meant to approach this. Firstly, I think of the fact that, as $E/F$ is Galois, $|Gal(E/F)|=|E:F|=|E:K||K:F|=|E:K|2$. Then, I know that an intermediate field of a Galois ...
cable's user avatar
  • 150
0 votes
0 answers
72 views

$E = \mathbb{Q}( \sqrt{p_1}, . . . , \sqrt{p_n})$. Show that $Gal(E/\mathbb{Q}) \cong (\mathbb{Z}/2\mathbb{Z})^n$

Hey I want to check if my solutions for this exercise are right. Let $E = \mathbb{Q}( \sqrt{p_1}, . . . , \sqrt{p_n})$ with $n$ pairwise different prime numbers $p_1, . . . , p_n$. Show that $E/\...
Marco Di Giacomo's user avatar
1 vote
1 answer
101 views

$Gal(LM/K)$ isomorphic to $ \{ (\sigma , \tau) \in Gal(L/K) \times Gal(M/K) : \sigma |_{L \cap M} = \tau |_{L \cap M} \} $

Let L/K, M/K be finite Galois extensions contained in some common field extension $\mathbb{K}$ of $K$, so we can speak about $L \cap M$ and the composite $LM$ , which is defined to be the smallest ...
Pch's user avatar
  • 13
-1 votes
1 answer
68 views

Some exercises about Galois Groups and Automorphisms of a finite field extension

Hey I have these two exercises where I am having some problems solving them: Let $E/K$ be a finite field extension, let $G$ be a subgroup of $Aut(E/K)$. Show: a) $G$ is finite and $|G|$ is a divisor ...
Marco Di Giacomo's user avatar
0 votes
0 answers
40 views

Exercises about Galois theory and isomorphism type of the Galois-Group [duplicate]

I have this two exercise where I have some problems: (a) We consider the finite field extension $E := \mathbb{Q}( \sqrt[4]{2}, i)$ over $\mathbb{Q}$. Show that $E$ is a Galois extension of $\mathbb{Q}$...
Marco Di Giacomo's user avatar
1 vote
1 answer
73 views

Extending a field homomorphism on a finite field

Let $p$ be a prime number. Let $a,b,c$ be positive integers, $a$ divides $b$ and $b$ divides $c$. Fix homomorphisms $f\colon\mathbb{F}_{p^{a}}\to\mathbb{F}_{p^{b}}$ and $g\colon\mathbb{F}_{p^{a}}\to\...
Object's user avatar
  • 339
0 votes
0 answers
45 views

I ask about the galois extension proof of the fundamental theorem

I was reading the book The Theory of Fields and Galois by J.S. Milne, and I have a question about a demonstration that is done in the book, in THEOREM 3.16 (FUNDAMENTAL THEOREM OF GALOIS THEORY) in ...
ruka's user avatar
  • 118
2 votes
1 answer
80 views

Is there an example, such that $\omega\in F(\beta)\land \omega\notin F$?

$F$ is perfect, $E=F(\beta)$, where $\beta^n=a\in F$, then $E$ is a Galois extension over $F$ if and only if the field contains the n-th primitive root $\omega$, where $\omega^n=1$ Question: I am ...
GGplay's user avatar
  • 181
1 vote
0 answers
88 views

$\Bbb Q \subseteq \Bbb Q(\sqrt {a + \sqrt b})$ is normal iff at least one of $a^2 − b$ and $(a^2 − b)/b$ is a square in $ \Bbb Q$

Let $a, b$ be integers such that $b$ is not a square. Show that the field extension $\Bbb Q \subseteq \Bbb Q(\sqrt {a + \sqrt b})$ is normal if and only if at least one of $a^2 − b$ and $(a^2 − b)/b$ ...
darkside's user avatar
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