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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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 ...
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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 ...
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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 ...
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Relatively Prime field extensions: Minimal polynomial coefficients

Premise: We are given field extensions $L/K$ and $L(a)/K$ such that degrees $[L:K]$ and $[K(a):K]$ are relatively prime. Question: I would like to show that the minimal polynomial of a over $L$ has ...
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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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If $k[a_1,a_2,…,a_n]=k(a_1,a_2,…,a_n)$ show that $a_1,…,a_n$ are algebraic over $k$.

I am trying to prove the following statment and need some help. Let $k$ and $E$ be fields such that $k \subset E$ and $a_1,a_2, \ldots ,a_n \in E$, if $k[a_1,a_2,...,a_n]=k(a_1,a_2,...,a_n)$ show ...
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+200

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Car}(F)=0$, $E/F$ 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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Cubic field extension question

Is the following statement true ? $$ 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}).$$
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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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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 ...
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checking the solution of problem in field extensions

here's the problem: suppose that $K|F$ is a field extension and $a\in K$ is an algebraic element over $F$ that it's minimal polynomial has odd degree.prove that $F(a)=F(a^2)$. i think this is a ...
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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 ...
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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 ...
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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 $\mathbb{Q}[x]$: ...
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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 ...
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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 $[K : k] = m$ ...
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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 ...
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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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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 ...
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Galois Extensions

I have to solve the following exercise: Let $f(x)=3x^4-3x^2+1$ be a polynomial in $\mathbb{Q}[x]$ and let $\alpha$ be one of its roots. 1) Determine $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and ...
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$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ 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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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 ...
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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 ...
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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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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 ...
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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 ...
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algebraic extension fields & intermediate subrings

here is the problem: let $E|K$ be an extension field,prove that it is an algebraic one if & only if every subring $R$ of $E$ containing $K$, is a field. i know "only if" side of proof. but i ...
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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 ...
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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 ...
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Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
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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 ...
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Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
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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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Find Galois Group

I want to know how to find a polynomial $f(x)$ of degree $6$ in $\mathbb{Q}[x]$ with Galois groups $G_f=\mathbb{S}_6$. I have a criterion to find a polynomial wich Galois group is $\mathbb{S}_p$ with ...
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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. ...
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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 ...
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independence of Galois conjugates

Let $L \supset K$ be a transcendental extension of fields, $K$ algebraically closed and let $y \in L'$ be algebraic over $L$, and let $\sigma(y)$ be its Galois conjugate for some $\sigma \in ...
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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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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 ...
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Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
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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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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 ?
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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 ...
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1answer
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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 $3$ ...
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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 ...
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Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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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 $L$. ...
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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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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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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 ...