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 ...

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Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$

Let $K$ be a field and $\mu_n$ a primitive $n$-th root of unity. Then I can embed $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$. For $K=\mathbb{Q}$ there would be even an ...
23
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2answers
1k views

Is every group a Galois group?

It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following ...
11
votes
1answer
274 views

Finite extensions of rational functions

I know that finite extensions of $\mathbb{C}(x)$ correspond to finite branched covers of $\mathbb{P}^1$, and this leads to an abstract characterization of the absolute Galois group of $\mathbb{C}(x)$ ...
3
votes
2answers
497 views

A polynomial whose Galois group is $D_8$

I need to construct such a polynomial, and more generally: given a group $G$, how can it be realized as a Galois group?
3
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4answers
186 views

How to prove $t^{23}+1$ irreducible in $F_p$?

I have tried to prove that $t^2+1$ is irreducible over $F_3$ by supposing to the contrary $t^2+1=(t+\alpha)(t+\beta)=t^2+(\alpha+\beta)t+\alpha\beta$. Then, $\alpha+\beta\equiv 0 \pmod 3, ...
3
votes
2answers
303 views

An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
2
votes
1answer
303 views

Galois extension and Tensor product

The following theorem is proved in Bourbaki's algebra. They use the technique of Galois descent. I'd like to know the proof without using it if any. Theorem Let $K$ be a field. Let $\Omega/K$ be an ...
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0answers
51 views

In a quadratic extension, does the orthogonal complement of trace have a nice name?

The trace of an element in a quadratic extension is the sum of it with its conjugate. Is there a name for the difference between it and its conjugate?
2
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1answer
244 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
1
vote
1answer
63 views

What is this quadratic form as a invariant of Galois Extensions?

Suppose that $E/F$ is a Galois extension and viewing E as a vector space over $F$, then quadratic from $Tr_F^{E}(\alpha^2)(\alpha\in E)$ carries some information of the extension. My question is that, ...
8
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2answers
344 views

How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?

From Berkeley Problems in Mathematics, Spring 1999, Problem 17. Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\ge 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in ...
2
votes
2answers
208 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
2
votes
1answer
99 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
2
votes
1answer
176 views

Integral Galois Extensions - Proposition 2.4 (Lang)

My question refers to the proof of Proposition 2.4, p. 341 in Lang's Algebra. Here is the context: Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a ...
3
votes
2answers
479 views

If $N$ is a normal subgroup of $G$ with $N$ and $G/N$ solvable, prove that $G$ is solvable

I'm trying to prove that if $N$ is a normal subgroup of $G$, with $N$ and $G/N$ solvable, then $G$ is solvable. Proving that $G/N$ is abelian would of course suffice, but I'm not sure if that's a ...
2
votes
1answer
200 views

Integral Galois Extensions (Lang)

I have trouble understanding an argument in the proof of Proposition 2.5, p. 342, of Lang's Algebra. The setup of my question is the following: Let $A$ be integrally closed in its quotient field $K$ ...
4
votes
1answer
324 views

A formula for the roots of a solvable polynomial

Let $F$ be a field and $p(x)\in F[x]$ a separable polynomial, denote $K$ as the splitting field of $p$ and assume that $K/F$ is Galois with a solvable Galois group. I don't understand if this imply ...
2
votes
2answers
154 views

Is a polynomial solvable by roots iff every irreducible factor is?

Let $F$ be a field, I asked myself if $p(x)\in F[x]$ is solvable by radicals iff every irreducible factor is solvable by radicals. My thoughts: If every irreducible factor is solvable by roots then ...
3
votes
1answer
308 views

The order of the Galois group of a cyclotomic field over a finite prime field [duplicate]

Possible Duplicate: For what $(n,k)$ there exists a polynomial $p(x) \in F_2\[x\]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$? Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} ...
3
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1answer
566 views

Subgroup of Galois group of polynomial over $\mathbb{Q}$

Let $K$ be the splitting field of $x^5-3 \in \mathbb{Q}[x]$. We can see $K = \mathbb{Q}(3^{1/5}, \zeta_5)$ where $\zeta_5 = e^{2 \pi i/5}$, and $[K: \mathbb{Q}] = 20$. It's easy to see $\sqrt{5} \in ...
3
votes
2answers
339 views

Proving there are no subfields

I am trying to solve Q11 at pg. 582 from the book Abstract algebra by Dummit and Foote, the question is: Let $f\in\mathbb{Z}[x]$ be an irreducible quartic whose splitting field has Galois group ...
6
votes
2answers
153 views

Showing $\mathbb{Q}(\sqrt[4]{2},i)=\mathbb{Q}(\sqrt[4]{2}+i)$ using the Galois orbit of $\sqrt[4]{2} + i$

The following is a problem from I. Martin Isaac's Algebra. Let $E=\mathbb{Q}(\sqrt[4]{2}+i)$. I am trying to show $\mathbb{Q}(\sqrt[4]{2},i)=E$ with the following hint: Find at least five ...
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2answers
222 views

Non-Galois cubic extensions

Is there a necessary and sufficient condition for when a cubic extension of $\mathbb{Q}$ is not a Galois extension?
2
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1answer
218 views

Transition between field representation

In a number of papers related to efficient implementation of the AES Sbox, people are computing stuff (the multiplicative inverse for instance) in GF(($2^4$)$^2$) instead of GF($2^8$). In some cases ...
6
votes
1answer
246 views

what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?

John Conway proved in his book, On Numbers and Games (ch6, theorem 49) that the set of all ordinals smaller than $\omega^{\omega^\omega}$ form a field of characteristic 2 that is isomorphic to ...
6
votes
1answer
605 views

Calculating the norm of an element in a field extension.

Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element ...
8
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2answers
2k views

Constructing a Galois extension field with Galois group $S_n$

Constructing a Galois extension field $E$ with $Gal(E/F)= S_n$ How do I construct one?
0
votes
1answer
270 views

How to prove with Galois theory that a cube root is not geometrically constructible?

I could not find any proof on the Internet. I am looking for a formal proof with an explanation for the uninitiated (my knowledge of Galois theory is very basic). With geometrically constructible I ...
4
votes
1answer
639 views

Infinite extension in Galois theory

I got this question while I am studying Galois correspondence. Let $K/F$ be an infinite extension and $G = \mathrm{Aut}(K/F)$. Let $H$ be a subgroup of $G$ with finite index and $K^H$ be the fixed ...
2
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1answer
108 views

Common factors of the ideals $(x - \zeta_p^k)$, $x \in \mathbb Z$, in $\mathbb Z[\zeta_p]$

I'm trying to understand a proof of the following Lemma (regarding Catalan's conjecture): Lemma: Let $x\in \mathbb{Z}$, $2<q\not=p>2$ prime, ...
1
vote
2answers
369 views

smallest subfield containing transcendental or algebraic element

Consider the extension $\mathbb R/ \mathbb Q$. what is the smallest subfield $\mathbb Q(\pi)$ of $\mathbb R$ containing $\mathbb Q$ and $\pi$. I think it is $\mathbb R$. A more general question: for ...
7
votes
1answer
160 views

If $f(x) \in F[x]$ is a polynomial solvable by radicals, does this imply that its Galois group is solvable when $F$ has characteristic $p > 0$?

We know the following theorem from Galois theory: Let $F$ be a field of characteristic $0$ and $f(x) \in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is ...
5
votes
1answer
148 views

Is $i\notin \mathbb{Q}(\zeta_p)$ for all odd primes $p$?

My main question is the title: for an odd prime $p$, denote a primitive $p^{\text{th}}$ root of unity by $\zeta_p$. Is it true that $i$ is not contained in the cyclotomic extension ...
1
vote
1answer
120 views

Lifting additive characters

Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$. Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive ...
0
votes
1answer
151 views

Intersection of compositum of fields with another field 2

The following is a previous question with an additional hypothesis: Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1\cap K=F_2 \cap K=M$, the extensions $F_i/(F_i \cap K)$ are ...
1
vote
1answer
130 views

Intersection of compositum of fields with another field

Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1 \cap K = F_2 \cap K = M$, the extensions $F_i / (F_i \cap K)$ are Galois, and $[F_1 \cap F_2 : M ]$ is finite. Then is $[F_1 F_2 ...
5
votes
2answers
455 views

Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$

Is there a clever way to determine a primitive element of the finite extension $$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$ On simpler examples, I've been able to find one by ...
27
votes
2answers
2k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
9
votes
3answers
2k views

Galois Group over Finite Field

I am having a bit of difficulty trying to answer the following question: What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$? So far I have factored $X^8-1$ as ...
4
votes
3answers
475 views

Proving the Möbius formula for cyclotomic polynomials

We want to prove that $$ \Phi_n(x) = \prod_{d|n} \left( x^{\frac{n}{d}} - 1 \right)^{\mu(d)} $$ where $\Phi_n(x)$ in the n-th cyclotomic polynomial and $\mu(d)$ is the Möbius function defined on the ...
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2answers
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Determine the Galois group of the splitting field of $(x^3-1)(x^2-5)$ over $\mathbb{Q}$

Determine the Galois group of the splitting field of $(x^3-1)(x^2-5)$ over $\mathbb{Q}$ I've been struggling with some of these Galois group questions.
0
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1answer
46 views

probability inequality resolution finite field

I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a ...
4
votes
3answers
470 views

Field Extensions as $F$ adjoin some element

Let $F$ be a field and $E$ an extension of $F$. Is it always possible to write $E=F(\alpha_1,\alpha_2,\ldots)$? If $E$ is a finite extension then I think it is possible to write ...
3
votes
2answers
456 views

Unramified p-adic extension implies Galois

I am looking for a short proof that if $L \supset K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$, then if $L/K$ is unramified, $L/K$ is Galois. I think the proof is related to somehow ...
4
votes
1answer
234 views

Number of embeddings in algebraic closure

I'm having trouble following the details of discussion on pages 9 and 10 of Neukirch's algebraic number theory book. Suppose $L$ is a separable extension of $K$ with degree n. Consider the set of ...
6
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3answers
747 views

Galois group of $x^6 + 3$ isomorphic to a copy of $S_3$ inside $S_6$

I have seen the the thread here related to the computation of the Galois group of the same polynomial. However, my question is not about the computation itself but about the group presentation of the ...
2
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2answers
508 views

A tower of normal extensions is normal

I am reading Kaplansky's notes on Galois theory. He defines a normal extension as follows: Let $M$ be any field and $K$ any subfield. $M$ is normal over $K$ if for any $u\in M$ but not in $K$, there ...
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2answers
628 views

Prove that the intersection of all subfields of the reals is the rationals

I'm reading through Abstract Algebra by Hungerford and he makes the remark that the intersection of all subfields of the real numbers is the rational numbers. Despite considerable deliberation, I'm ...
0
votes
1answer
115 views

Why is the following map well defined?

Let $H\leq G=\operatorname{Gal}(K/F)$ ($K/F$ is a finite galois extension), why is the following map well defined: $\varphi:G/H\to\Gamma_F(K^H,K)$ defined by $\sigma H\mapsto\sigma|_{K^H}$ ,where ...
4
votes
2answers
230 views

$p$-Sylow subgroups of a group of order $5^3\cdot 29^2$

I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup ...