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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Prove $K$ is an extension by radicals

Let $K$ be the splitting field of $x^3+x^2+1\in \mathbb{Z}_2[x].$ I'm trying to show that $K$ is an extension by radicals. I know that if $K$ is an extension by radicals, then there exists a chain of ...
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Find $Gal(K/\mathbb{Q})$ and show that $K/\mathbb{Q}$ is normal where $K=\mathbb{Q}(a)$

Let $K=\mathbb{Q}(a)$ and $a$ is a root of $x^3+x^2-2x-1 \in \mathbb{Q}[x]$. Find $Gal(K/\mathbb{Q})$ and prove that $K/\mathbb{Q}$ is normal. I just noticed that $a^2-2$ is also a root of the ...
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Show that this polynomial (involving n-th roots of unity) has rational coefficients

I am tasked with showing that the polynomial $\Phi(x) = \prod_{i=0}^{\phi(n)} (x - \omega_i)$ contains only rational coefficients. To give you an example, if $n = 10$, the polynomial is just ...
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Can we describe nicely all the rational numbers of the form $x^{2}-xy+y^{2}$?

Let $\zeta$ be a primitive cubic root of unity, i.e., $\zeta$ is a complex number such that $\zeta^{3}=1$ and $\zeta\neq1$. Let us consider the Galois extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ (the ...
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Counter-examples of Galois Correspondence

What are some examples of a separable field extension $L/K$ and a subgroup $H$ of $\text{Aut}(L/K)$ such that $\text{Aut}(L/L^H) \neq H$? Here $L^H$ means the fixed field of $H$.
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Proof Enquiry, Field of order $p^n$ [duplicate]

I want to prove that there exists an inclusion $\mathbb{F}_{p^a} \hookrightarrow \mathbb{F}_{p^b}$ iff $a \vert b$. Suppose that $a \vert b$, then $b =ac$ for some $c \in \mathbb{Z}$. Consider then ...
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For each polynomial $p \in K[t]$, there exists another polynomial $g$ such that $\{p(\epsilon_i): 1\leq i \leq n\}$ is its set of roots

I'd like to solve the following problem: Let $K$ be a field, $f \in K[t]$ a polynomial of degree $n$ and $E$ a splitting field of $f$ over $K$ in which $f$ has $n$ distinct roots ...
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Fundamental theorem of Galois Theory problem

Let $E/F$ be a Galois extension of degree $p^k$. Prove that there exists an intermediate field $K$ with $[E:K] = p$ and $K/F$ Galois of degree $p ^{k−1}$. I think I know how to prove the former but I ...
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Frobenius maps and irreducible functions on finite fields

Let $\mathbb{F}_q$ be a finite field of order $q=p^n$ for some prime $p$ and $n>1$. Suppose both $f(x)=x^2-ax+b$ and $g(x)=x^2-a'x+b'$ are both irreducible. If, assuming that either $a=a'=0$ or ...
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Calculating the Matrix of a Transformation Using Bases of Field Extensions

I'm trying to understand this topic in my Abstract Algebra class: Suppose that we have a finite field extension $L/F$ and let us choose $a \in L$. We'll define the transformation $T_{a} : L \to L$ ...
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Galois field of order 2 constituting a Boolean algebra

We know that the the set $\{0,1\}$ constitutes a Boolean Algebra over the usual $OR$ and $AND$ operations. However, because of the lack of an additive inverse for $1$ this does not produce a Galois ...
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Find a primitive element in the splitting field of $x^4-8x^2+15$.

I'm trying to find a primitive element in the splitting field of $x^4-8x^2+15$. I don't know in general what should I do with this kind of questions. Should I solve the roots and get the Galois group? ...
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Find the Galois group of $x^4-x^2-6$.

I'm trying to find the Galois group of $x^4-x^2-6$. I think there are 4 roots, thus I guess the Galois group is $A_4$? But I don't know in general how to solve this.
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Field extensions that decompose into towers of degree$\leq n$ extensions

Let $F$ be a field and let $n$ be a natural number. Consider the class of field extensions $E/F$ that decompose into towers $E=E_k/E_{k-1}/\cdots/E_1/E_0=F$ such that $[E_{i+1}:E_i]\leq n$ for ...
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Find the Galois group of $x^3-5$ over $\mathbb{Q}$.

In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, ...
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Field theory: an equality involving the number of homomorphisms from an extension $E$ of $F$ to $\overline{F}$

First some notation. Let $F$ be a field, $E$ an algebraic extension of $F$ and $\overline{F}$ the algebraic closure of $F$. Let $\{E:F\}$ represents the number of non-zero homomorphisms from $E$ to ...
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Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (i.e. for profinite groups?) I suspect that the theory of Galois ...
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26 views

No $p$-th root implies $X^{p^n}-a$ irreducible for all $n \in \mathbb{N}$

While doing exercises of Chapter IV in Lang's algebra, I encountered the following problem: Suppose char $K=p$. Let $a \in K$. If $a$ has no $p$-th root in $K$, show that $X^{p^n}-a$ is ...
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Galois group of polynomials

Let $f$ be an irreducible polynomial over a field K, and $\deg f = 4$, with roots $a,b,c,d$. Let $g$ be a cubic resolvent with roots $\alpha,\beta,\gamma$. And $\alpha=ab+cd, \beta=ac+bd, ...
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Showing normalizer of Galois group

Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove ...
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Showing that the group of $n^{th}$ roots of unity form a group

I know they are $\{1, \gamma, \gamma^2, ..., \gamma^{n-1}\}$, where $\gamma$ is the primitive n-th root of unity. To show that this is a group (the four axioms), I did this: Since $\gamma^i$ is a ...
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for any $k \in \mathbb{Z}^{+}$, how do we know $\omega^1$ is a primitive root of $x^k - 1$?

I know that it is definitely a root. The set $\{1, \omega, ..., \omega^{k-1}\}$ are the roots of unity. But how are we guaranteed that it even exists and that furthermore it is a primitive root? ...
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How is $Gal(\mathbb{Q}(\sqrt[4]{3}, i)$ equal to $D_8$? (dihedral group of order 8)

From what I gathered, the automorphisms are as follows: $\iota$ (the identity map) $\sigma_2: \sqrt[4]{3} \rightarrow \sqrt[4]{9}$ (squaring map) $\sigma_3: \sqrt[4]{3} \rightarrow \sqrt[4]{27}$ ...
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Given a tower of field extensions, does this equality involving Galois group orders hold in general?

Suppose we have a tower of field extensions: $\overline{F} \subset K \subset E \subset F$ Is it true in general that $|G(K/F)| = |G(K/E)| \cdot |G(E/F)|$? I was able to verify some specific ...
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Showing that the multiplicative group of roots of $x^n - 1$ in $\overline{F}$ has $\phi(n)$ distinct generators

Basically the question is how can it be shown that the n-th cyclotomic extension, which is a cyclic group under multiplication, has exactly $\phi(n)$ distinct elements, where $\phi$ is Euler's totient ...
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Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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polynomial in fixed field

Please suggest a method to solve this? Let $E$ be a field and let $H$ be a subgroup of $Aut(E)$. Let $F$ be the fixed field of $E$ under $H$ and let $f(X) ∈ E[X]$ be a monic polynomial that splits in ...
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Field extension if element fixed by only identity [closed]

Suppose that $E$ is a Galois extension of $F$ and that $α \in E$ is left fixed by only the identity in $\text{Gal}(E/F)$. Prove that $E = F (α)$. Please suggest how I should proceed. Thanks!
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Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
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Computing Galois groups of function fields in sage

I found documentation on how to compute galois groups for number fields in sage. Is it possible to do the same for function field extensions? I only need it in the simple case of $t - f(x)$ over ...
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Showing that this polynomial is in F[x] (Galois theory, given K is some extension of F)

Suppose we have some extension $K$ of some field $F$ and the Galois group $G(K/F)$ which consists of automorphisms $\sigma$ of $K$. How could we show the following: For any $a \in K$, show that ...
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(Differential Galois Theory) Where is the proof that the three-body-problem is unsolvable?

I'm looking for a proof, which shows that "the 3-body-problem" in physics is mathematically unsolvable. Does anyone know some URLs that contain a proof in mathematical detail? You know, in ...
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On the question of the Galois group of some polynomial. [duplicate]

I want to ask you some question on the Galois group of some polynomial. Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers. Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in ...
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How to show that $\mathbb{Q}(\sqrt[6]{2}, i)$ is a splitting field over $\mathbb{Q}$

That is, find a minimal polynomial that splits over this field. So, is it enough to do this?: $x = \sqrt[6]{2} = 0$, therefore $x^6 = 2$ and thus $x^6-2$ is a factor. $x = i$, therefore $x^2 = ...
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1answer
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The Galois group of a polynomial over a field and over some extension (updated)

Let $f(x)\in F[ x]$ and $K/F$ be a field extension. Show that the Galois group of $f(x)$ over $K$ is isomorphic to a subgroup of the Galois group of $f(x)$ over $F$. Let $E$ be the splitting ...
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1answer
21 views

Show that deg $P$ = |Gal $(L/K)$|.

Let $K \subset L$ be a finite Galois extension. Assume that there is $a \in L$ such that $\sigma(a) \not= a$ for every $\sigma \in$ Gal$(L/K)$. Let $P$ be the minimal polynomial of $a$ over $K$. (a) ...
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Subfield of $\mathbb{Q}(\sqrt[n]{a})$

Exercise 14.7.4 from Dummit and Foote Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in \mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$(i.e., $x^n-a$ is irreducible). Let $E$ be any subfield of ...
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How to find the subfields fixed by subgroups of $\text{Gal}(x^3 - 4x + 2)$?

I'm trying to draw the subgroup diagram for the Galois group of $x^3 - 4x + 2$, and find the subfield fixed by each subgroup. Here's what I know: the group is $S_3$. I solved the equation through ...
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What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
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Minimal polynomial of a primitive element for Galois extensions with Galois group $S_n$

Let $K$ be a global field, $f(x)\in K[x]$ be an irreducible separable polynomial and $L$ be the splitting field of $f(x)$. Suppose that the Galois group of $L$ over $K$ is the symmetric group ...
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1answer
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About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n $ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n $ ( A subgroup $G$ ...
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Finding the fixed subfield (Galois theory)

Let's say we are working with the field extension $\mathbb{Q}(\gamma)$, where $\gamma$ is the seventh root of unity. I know my basis for this extension will thus be: $\{1, \gamma, \gamma^2, ...
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Some natural question on subfield of Galois extension

Let $\alpha,\beta\in \mathbb{\overline{Q}}$ and assume $\deg(\text{Irr}(\alpha,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q(\alpha)})=2$. Then I strongly guess that ...
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1answer
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Finding the fixed subfield corresponding to a cyclic subgroup of the Galois group

Let's say I have a field extension $E$ of some field $F$ and I also know the Galois group of $E$ over $F$. Suppose I have a subset of this Galois group which is cyclic, thus generated by some ...
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1answer
45 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is ...
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39 views

Show that any group of order $2p$ is soluble

Let $|G| = 2p$. The result is clear is $p$ is even. The proof goes on to show that there is only one subgroup of order $p$ is p is an odd prime - my question is that, why do we need to show that ...
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1answer
24 views

Prove that $F \subset \sigma L$ is also radical.

Suppose that we have finite extensions $F \subset L \subset M$ and $\sigma \in Gal(M/F)$ and assume that $F \subset L$ is radical. Prove that $F \subset \sigma L$ is also radical. Since the ...
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1answer
37 views

All intermediate sub extensions of $\mathbb{Q} \subseteq K \subseteq \mathbb{Q}(\zeta_8)$.

I know there is a similar question posted on Stack Exchange, however it deals with periods, and I do not understand the solutions provided. I know that the Galois Group of the field extension ...
2
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1answer
42 views

Extension of field homomorphisms and pullback square

Let $E/k$ and $F/k$ be two subextension of a field extension $K/k$. The following square induced by restriction functions is always pullback square (in category of sets and functions)? ...
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Intersection of two subfields of $F(X)$ [duplicate]

Let $E=F(x)$ for a field $F$ of characteristic $0$. Show that $F(x^2) \cap F(x^2-x) = F$ as subfields of $F(x)$. I could use a hand with this... Thanks