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 (galois-...

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1answer
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Galois group of $x^{2^k}+1$

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?
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2answers
88 views

monomorphism on an algebraic field extension

let $E|K$ be an algebraic extension and $\phi:E\rightarrow E$ a $K $-algebra monomorphism,prove that $\phi$ is onto. i assume $\alpha\in E-\phi(E)$ to make a contradiction and i assume $f(x)$ to be ...
1
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1answer
80 views

Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
4
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1answer
44 views

product considering the period of index over a cyclotomic extension

This is an exercise from Milne Galois Theory (Chapter 3 Exercise 13). Let p be an odd prime, and let $\zeta$ be a primitive $p^{\text{th}}$ root of 1 in $\mathbb C$. Let $E = \mathbb Q[\zeta ]$, and ...
0
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1answer
60 views

Prove that this extension is Galois with galois group the quaternions

Let $M=\mathbb Q(\sqrt{2},\sqrt{3})$ and let $E=M(\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)})$. Prove that: M is Galois over $\mathbb{Q}$ Show that $E$ is Galois over $\mathbb Q$ with Galois group the ...
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0answers
37 views

galois group of a biquadratic involving primes.

Let $f(x)=x^4-px^2+q \in \mathbb Q[x]$ be a polynomial with $p,q$ be distinct primes. Prove that $f$ it's irreducible over $\mathbb Q$. Prove that it's Galois group is the dihedral. I proved the ...
2
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0answers
303 views

Example of a non-separable normal extension

I'm trying to give an example of a normal field extension $K|F$ that is not separable. I now that if $F$ is finite or char$(F)=0$, $K|F$ is automatically separable, thus, I must look into infinite ...
3
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1answer
49 views

different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If $\...
17
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4answers
370 views

Value in retracing mathematicians' steps (specifically Galois)?

So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus ...
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2answers
33 views

Cubic field extension question

Is the following statement true ? $$ 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}).$$
2
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1answer
116 views

Showing $\operatorname{Aut}(\mathbb{R})$ is abelian

I have an exercise (not for a class) that asks whether $\operatorname{Aut}(\mathbb{R})$ (field automorphisms) is abelian. I know that $\operatorname{Aut}(\mathbb{R})$ is just the trivial group, but ...
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1answer
75 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
3
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1answer
82 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
3
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2answers
64 views

Calculating a minimal polynomial over $\mathbb{Q}(\sqrt[3]{2})$

Let $L_1=\mathbb{Q}(\omega\sqrt[3]{2})$ where $\omega=e^\frac{2\pi i}{3}$ and $L_2=\mathbb{Q}(\sqrt[3]{2})$. I want to calculate $[L_1L_2:L_2]$, that it is the degree of the minimal polynomial over $...
1
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2answers
162 views

Calculate the degree of the field extensions.

I have been staring at this question for a while. I'm sure there is a little trick I am missing...anyway, it is the following: $ f = x^3 + x + 3 $ a) Show $f$ is irreducible over $\mathbb{Q}[x]$: ...
5
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2answers
156 views

Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
2
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1answer
135 views

Some Galois theory

I have a question on field extensions, and I can't seem to find precise answers when browsing through online notes etc. Here it is: suppose $K$ and $k$ are fields with $k \leq K$ and $[K : k] = m$ ...
4
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2answers
76 views

Irreducible factors of a polynomial in a Galois extension

Let $E|F$ be a finite Galois extension and $f(x) \in F[x]$ an irreducible polynomial. Prove that each of the irreducible factors of $f(x)$ in $E[x]$ have the same degree. An idea: Let $\phi \in Gal(E|...
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2answers
56 views

Galois subextensions in a Galois extension

Let $F \subset E \subset L$ be fields such that $L/E$ and $E/F$ are both Galois extensions. Is $L/F$ necessarily a Galois extension?
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0answers
333 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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2answers
41 views

Field extensions and irreducibility

I'm having trouble trying to show that the function f=x^3 + x + 3 is irreducible in the rationals. I tried using Eisensteins criterion but it didn't work as it doesnt satisfy all conditions. the ...
1
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1answer
38 views

$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
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0answers
73 views

Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that $K$...
2
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1answer
156 views

Non-Galois number fields and complex embeddings

Let $K$ be a number field. $K$ is a normal extension of $\mathbb{Q}$ iff $\exists f(x)\in\mathbb{Q}[x]: K$ is the splitting field for $f(x)$. A field extension is Galois iff it is normal and separable,...
3
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3answers
32 views

Let $G = \text {Gal}(L/K)$. How do I see that $\phi \in G$ is completely determined by $\phi\mathbb( \sqrt[4] {3})$ and $\phi(i)$?

Let $K = \mathbb Q$ and $L = \mathbb( \sqrt[4] {3}, i)$. I see that $L \supset K$ is a an Galois extension. Also, after computations I've $[L : K] = 8$ (degree of $L$ over $K$ considered as a vector ...
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0answers
158 views

Deriving the Resolvent Cubic From Elementary Symmetric Functions

On page 614 of Dummit and Foote's Abstract Algebra, while deriving the resolvent cubic for a quartic $g$, they go from a trio of elements $\theta_i, i=1,2,3$ which were defined in terms of the roots ...
2
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1answer
68 views

relation between the characteristic polynomial and the minimal polynomial

Define $l_a : F(a) → F(a) $ by $ l_a(x)=ax$, when $[F(a):F]=n$ . show that the minimal polynomial of $a$ over $F$ is the same as the minimum polynomial of $l_a$ as defined in linear algebra. this ...
1
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1answer
33 views

a question about field extensions and tower formula

if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction ...
4
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1answer
117 views

On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
2
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1answer
50 views

an exercise in Galois theory about polynomials

find a field $F$ and different polynomials $f(X),g(X)\in F[X]$ that for every $\alpha \in F$ we have $f(\alpha)=g(\alpha)$. prove that it is impossible if $F$ is infinite. i think this example works:...
5
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1answer
225 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
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2answers
44 views

Algebraic Extensions

I have the following question: there is this statement i can't understand: Let $A$ be an integral domain which is integrally closed ( in its field of franctions ) and let $K$ be its fraction field. ...
3
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2answers
83 views

If $f(x)\in\mathbb{Q}[x]$ of degree $p$ and $\operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$ then $f(x)$ is irreducible.

Let $f(x)\in\mathbb{Q}[x]$ , $p$ prime, $\deg f(x)=p$ and $G = \operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$, where $K$ the the splitting field of $f(x)$ over $\mathbb{Q}$. Show that $f(...
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0answers
52 views

Finite characteristic splitting fields of low degree polynomials

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. Let $f_p(x) \in \mathbb{Z}_p[x]$ denote the polynomial $f \bmod{p}$ (where $p$ is a prime). We say $p$ is good for $f$ if $f_p(x)$ splits (into linear ...
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2answers
41 views

If $\deg(f) > p^k$ then $f$ as an irreducible divisor of degree $> k$

Let $p$ be prime and let $f \in \mathbb{F}_p[X]$ with no repeated roots. Let $k \in \mathbb{N}^*$ such that $\deg(f) > p^k$. Show that $f$ has an irreducible divisor of degree $> k$. My work ...
2
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1answer
54 views

What is the difference between the algebraic function fields and the fields itself

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself. Thanks
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1answer
98 views

Sum of roots of unity an algebraic integer proof

Let S be the sum of a finite number of nth roots of unity (where n is fixed, and the sum is non-zero). How do I go about showing that S is an algebraic integer in the cyclotomic field of order n ?
2
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1answer
115 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
7
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0answers
162 views

How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of $\...
2
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1answer
94 views

Decomposition group of a prime ideal and root of polynomials

Let $f(x)$ be a monic irreducible polynomial with integer coefficient. Let $K$ be the splitting field of $f$ and $\alpha$ one of its roots. Let $p$ a prime number such that $p$ does not divide $disc(...
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2answers
138 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
4
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1answer
214 views

Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though $\...
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2answers
94 views

dimension of $\mathbb{Q}(\sqrt2)$

How can I prove that the splitting Field $\mathbb{Q}(\sqrt2)$ over the rational numbers $\mathbb{Q}$ is two dimensional vector space over $\mathbb{Q}$ ?
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1answer
44 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
5
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1answer
60 views

How to write a particular fixed field as a simple extension of $\mathbb{R}$? (Morandi's book)

I'm working in the following problem from Morandi's Book Field and Galois Theory: Let $A$ =$\left(\begin{array}{cc} a & b \\ c & d \end{array} \right)$ with $a=d=-1/2$ and $c=-b=\sqrt{3}...
2
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1answer
102 views

Normal extension and group of automorphisms

Let $K \supset F$ a normal extension (finite). Show in details that, for every $\alpha \in K$, the polynomial $$f(x)=\prod_{\sigma \in Aut_F(K)}(x-\sigma(\alpha))\in F[x];$$ My attempt: Since $K$ ...
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0answers
155 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
4
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3answers
68 views

$[F(t):F(t^n)]=n$ where $t$ is trascendental

Let $F$ be a field and let $t$ be trascendental over $F$. Prove that $[F(t):F(t^n)]=n$. Obviously $[F(t):F(t^n)]\le n$ since the polynomial $f(x)=x^n-t^n \in F(t^n)$ has $t$ as a root. But I don't ...
1
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1answer
42 views

How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series ...
1
vote
1answer
50 views

Prove that there exists a sequence of intermediate fields.

How can I prove that for $K\subset L$ - the Galois extension of degree $p^n$, where $p$ is prime, there exists a sequence $K=K_{0}\subset K_{1}\subset \cdots \subset K_{n}=L$ such that $[K_{i}:K_{i-1}]...