Questions tagged [galois-theory]
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-connections).
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How does the Galois group of a splitting field act on a specific element?
Let $charK \neq 2$, $L/K$ be the splitting field of a separable polynomial $f \in K[x]$.
Further let $a_1,...,a_n \in L$ be the roots of $f$.
How does the Galoisgroup $Gal(L/K)$ act on $c:=\prod_{1 \...
- 356
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Order of galois group of splitting field
Something silly question, but I can’t figure out what is wrong.
For example, suppose $K$ be a splitting field of $x^4 - 4x^2 -1$ over $Q$.
Of course, it is irreducible, and zeros are $\pm \sqrt{2+\...
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If $I=F$ , $K/S$ purely inseparable, $S/F$ separable then $K/I$ is not separable
In Patrick Morandi's book Field and Galois Theory Example 4.24 they write Let $k$ be a field of characteristic 2, let $F=k(x, y)$ and $S=F(u)$, where $u$ is a root of $t^2+t+x$, and let $K=S(\sqrt{u y}...
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Properties of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbits
Let $\alpha \in \overline{\mathbb{Q}}$ be an algebraic number so that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = N$, with $N$ a strictly positive integer. Then $A := \mathrm{Gal}(\overline{\mathbb{Q}}/\...
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Determining intermediate fields of the splitting field of $x^3-3\in \mathbb{Q}[X]$ over $\mathbb{Q}$
As an exercise I am trying to do the following:
Let $L$ be the splitting field of $x^3-3\in \mathbb{Q}[X]$ over $\mathbb{Q}$.
I want to determine $Gal(L/\mathbb{Q})$, all subgroups of $Gal(L/\mathbb{Q}...
- 356
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36
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Pove that the Galois group of $p(x) = f(x^2)$ is not abelian
Let $f(x) \in \mathbb Q[x]$ be an irreducible cubic polynomial with three real roots $\alpha, \beta, \gamma \in \mathbb R$ such that $\alpha$ is
negative and $\beta$ is positive. Prove that the Galois ...
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Is the Galois action on $k$-algebra homomorphisms transitive?
I am reading the proof of Proposition 2.19 in Liu's book on Algebraic Geometry and Arithmetic Curves. The proposition is as follows:
Let $X$ be an algebraic variety over $k$, and let $K / k$ be a ...
- 1,293
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49
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Galois Theory: What is the Underlying Significance of the Solvability Criterion
Alright everyone, here's the big Galois theory question. I see many people ask similar questions here, but mostly it seems it's people who do not already have much knowledge of Galois theory, so I ...
- 21
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Clarifying the meaning of insolvability of quintic
Insolvability of quintic says that no analogue of quadratic formula is available for quintic equations. For quadratic equations, the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives you all the ...
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Minimal Polynomial of $\sqrt[3]{2}$ over $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ and degree of field extension.
Hello and thanks in advance for any responses. I'm stuck on the following problem: "Show that the polynomial $X^3 - 2$ is irreducible over $\mathbb{Q}(\omega)$, where $\omega = e^{\frac{2\pi i}{3}...
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On the definition of algebraic closure
Let $F$ be a field. By definition, the following are equivalent:
$F$ is algebraically closed.
Every nonconstant polynomial in $F[x]$ splits over $F$.
Every nonconstant polynomial in $F[x]$ has a ...
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Is the Galois group of L/K a cyclic group where K, L are the 2^k, 2^l -th cyclotomic field with l > k > 2? [closed]
Assume $K, L$ are $2^k, 2^l$-th cyclotomic field, respectively, that is, $K = \mathbb{Q}[x]/(x^{2^{k-1}}+1), L = \mathbb{Q}[x]/(x^{2^{l-1}}+1), l > k > 2$. Then $Gal(K /\mathbb{Q}) \cong \mathbb{...
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Why are the algorithms for finding the Galois group of a polynomial generally flawed?
In Galois theory, the endeavor of algorithmically finding the Galois group of a polynomial $f$ over the field $\mathbb{Q}$, where $f\in\mathbb{Q}[z]$ is generally not well-accomplished. By this, i ...
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A question from Dummit-Foote on showing nontrivial algebraic extensions of a certain field (Section 14.9, Q.15)
I am trying to solve a problem from Dummit-Foote's Section 14.9 (problem 15):
Let $K_0= \mathbb Q$ and for $n > 0$ define the field $K_{n+1}$ as the
extension of $K_n$ obtained by adjoining to $...
- 1,779
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0
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40
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Compositum of Galois extension
Suppose $p,q$ are distinct primes. $K_1$ is a Galois extension of $\mathbb{Q}(\mu_{pq})$ of degree $p$, $K_1 \subseteq \mathbb{C}$. $K_2$ is a Galois extension of $\mathbb{Q}(\mu_{pq})$ of degree $q$, ...
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Specific example of fixed field using Artin's Theorem
I'm attempting a question as follows:
Let $K = \mathbb Q(x,y)$, where $x,y$ are independent transcendentals, and consider the group $G$ of automorphisms generated by
$$\sigma: \quad x \mapsto y,\quad ...
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A field has a unique algebraic extension of each degree if and only if its absolute Galois group is the profinite completion of integers
I would like to know how to prove that a field $K$ which has a unique algebraic extension of each degree has its absolute Galois group isomorphic to the profinite completion of the integers $\hat{\...
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39
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$K^{\text{quad}}$ and complex conjugation
Let $K\subset\mathbb{C}$ be a field that maps to itself under complex conjugation. Prove that $K^{\text{quad}}\subset\mathbb{C}$ also maps to itself under complex conjugation.
$K^{\text{quad}}$ is ...
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Number field, $K_i\neq K_{i+1}$
Let $K$ be a number field, prove: For the fields $K_i$, we have $K_i\neq K_{i+1}$ for all $i\geq 0$.
$K_i$ is defined as: $K_0=K$, and $K_i=K_{i-1}(\sqrt{K_{i-1}})$ for $i\geq 1$. Also, $K(\sqrt{K})$ ...
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How large is the gap in Ruffini's 1813 proof that there is no general quintic formula?
I'm reading Ruffini's final attempt at showing there is no general quintic formula which appeared in 1813 see here. (This is a much shorter proof than his first proof of 1799 in his Teoria generale ...
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How does $x^5-5$ factor over $\mathbb{F}_p$ for different values of $p$ mod 5?
For $p \neq 5$, I need to find the degrees of the factors of $f(x)=x^5-5$ in the cases where
a) $p \equiv 2$ or $3$ mod $5$ (show $f$ is the product of a linear factor and irreducible quartic over $\...
- 517
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34
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Finding fixed field of automorphism group (extension by transcendental elements)
Given $K=\mathbb{Q}(x,y)$ where $x$ and $y$ are independent transcendentals over $\mathbb{Q}$ and $G=\langle \sigma, \tau \rangle$ where $\sigma$ and $\tau$ are automorphisms of $K$ with $\sigma(x)=y$,...
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63
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Product of almost all Galois conjugates
I'm trying to prove the following: Given a matrix $M \in \mathbb{Z}^{n\times n}$ with an irreducible characteristic polynomial $f$ (irreducible over $\mathbb{Z}$ or $\mathbb{Q}$). If I'm not mistaken, ...
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Basic proofs on Weil groups
I had some questions regarding the Weil group. We defined a surjective map $res: \operatorname{Gal}(L/K) \rightarrow \operatorname{Gal}(k_L/k)$, where $k_L,k$ is the residue field of $L,K$ ...
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Example of a field on which every irreducible polynomial has degree a power of $p$
Exercise A-47 in Milne's Fields and Galois Theory notes asks to prove that if $p$ is a prime number and $F$ is a field of characteristic zero such that every irreducible polynomial $f(X)\in F[X]$ has ...
- 2,030
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Galois Theory: What, Exactly, is a General Polynomial?
I am learning Galois theory and am very close to a completed proof of the insolvability of the quintic. It remains to show that a polynomial can be constructed (over a reasonable field $F$) with ...
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How can I prove this regarding Cyclotomic Cosets?
For a positive integer $n$, let $[n]$ denotes the set $[n]:=\{1,2,3,...,n\}$.
Let $m$ be a positive integer, then define a set $A$ as
$$A:= [2^m - 2]\big\backslash [2^{m-1}-2]$$.
Now, define the ...
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Trace map of elliptic curve and its kernel
Let $E:y^2=x^3+17x$ be an elliptic curve over $\Bbb{Q}$.Let admit $E(\Bbb{Q})=\{(0,0),∞\}$ here.
Let‘s define a trace map $f:E(\Bbb{Q}(i))\to E(\Bbb{Q})$ defined by $P \mapsto P^{\sigma}+P$($\sigma$ ...
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How to rationalize the denominator $\frac{1}{1 + \sqrt[3]{5} - \sqrt[3]{25}}$
This is a review problem for an introductory Galois theory course.
Rationalize the denominator $\frac{1}{1 + \sqrt[3]{5} - \sqrt[3]{25}}$.
There could be many ways to do this, but it's implicit that ...
- 2,052
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Fixed field theorem
I'm stumbled upon a trivial detail of the proof of Fixed Field Theorem in Artin Algebra. Let $H$ be a finite group of automorphisms of a field $K$, and let $F = K^H$ be its fixed field. Then $K$ is a ...
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Profinite completion motivation
I have started galois theory recently, and curiosity quickly leads one to the subject of profinite groups. Although I have yet to be comfortable using these, I get what they are and we define them as ...
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Check for this Compute Galois groups of the following polynomials
Given polynomial $x^3 + t^2x− t^3$ over $k$, where $k = C(t)$ is the field of rational functions in one variable over complex numbers C.
This problem comes from Johns Hopkins University Spring 2010 ...
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If $K \subseteq \mathbb{C}$ is a Galois extension of $\mathbb{Q}$ and $\mathrm{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$, then $i \notin K$?
I want to prove that if $K \subseteq \mathbb{C}$ is a Galois extension of $\mathbb{Q}$ such that $\mathrm{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$, then $i \notin K$.
What I have tried: assume ...
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Examples where minimal polynomial is smaller over a simple extension.
I was working on some intuition-building exercises for Galois theory and ran into a problem that was a bit annoying. Are there any easy-to-check examples of algebraic numbers $a,b$ such that $[\mathbb{...
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Does imply f and g are relatively prime in K[x], that f- yg is irreducible in K(y)[x]?
Problem: Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in
$K[x]$. Show that $f- yg$ is irreducible in $K(y)[x]$.
My attempt: Consider the polynomial $f-gY\in (K[x])[Y]$. This ...
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Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the extension.
This is a follow-up to my previous question; I wanted to give a different example to make sure I understand what is going on.
Compute $[ \mathbb Q(\zeta_8) : \mathbb Q(i)]$ and find a basis for the ...
- 2,052
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Compute degree of $\mathbb Q(\zeta_8)$ over $\mathbb Q(\sqrt 2)$ and find a basis.
This is a review problem for a Galois theory course.
Compute $[\mathbb Q(\zeta_8): \mathbb Q(\sqrt 2)]$ and find a basis. (Where $\zeta_8$ is the primitive $8$th root of unity).
I compute the degree ...
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Galois group of $x^4 +3 \in \mathbb Q[x]$ using Kaplansky's Theorem
This is exercise 4.5 from Baker Galois Theory:
Use Kaplansky's theorem to find the Galois group of the splitting field $E$ of the polynomial $x^4 +3 \in \mathbb Q[x]$ over $\mathbb Q$. Determine all ...
- 2,052
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Reducibility of $f \in \mathbb{Q}(\alpha)[x]$ for $[\mathbb{Q}(\alpha):\mathbb{Q}] \geq 2$ and $f \in \mathbb{Q}[x]$ Irreducible?
I'm struggling to make any progress on the following problem.
Let $K = \mathbb{Q}(\alpha)$ be an algebraic extension of $\mathbb{Q}$ with $[\mathbb{Q}(\alpha):\mathbb{Q}]=m$, and suppose that $f \in \...
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Determine the galois group of this 6 degree polynomial over $\mathbb{Q}$
Determine the Galois group of $f(x)$ over $\mathbb{Q}$
$f(x)=x^6+22 x^5-9 x^4+12 x^3-37 x^2-29 x-15$
This question comes from Johns Hopkins University Fall 2018 algebra qualifying.
I have found it is ...
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Explicitly finding cyclic extensions over $\mathbb{Q}$
I'm working through some algebra qualifying exam questions, and came across the following problem which has totally stumped me.
Let $n \in \mathbb{N}$.
a) Explicitly construct a field extension $K_{/\...
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Residue field of the fixed field of the inertia group
This is from JS Milne's notes on algebraic number theory, the chart on page 139.
Let $L/K$ be a Galois extension of number fields, $p$ a prime ideal of $O_K$ and $\mathfrak{P}$ a prime ideal of $O_L$ ...
- 5,596
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25
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Primitive Elements and Bases of Fields [duplicate]
Good people!
I'm trying to prove a certain something, and I've reached a point where the whole thing will be complete if I can just prove the following lemma, which I'm actually not entirely certain ...
- 1,796
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21
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Showing that intermediate field $D$ is contained in intermediate subfield $E$ given these conditions.
Suppose $K/F$ is a Galois extension. Suppose $E$ and $D$ are intermediate fields of $K/F$. Let $p$ be a prime and assume that the following hold: (i) $[E:F]$ and $[D:F]$ are powers of $p$, (ii) $p$ ...
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1
answer
102
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Product vs. join of subgroups of a finite group
Hypotheses:
$G$ is a finite group
$A$ and $B$ are subgroups if $G$.
Definitions:
$AB = \{ ab \mathbin{|} a \in A, b \in B \}$ is the product of $A$ and $B$
$A \vee B$ is the join of (subgroup of $G$...
- 706
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1
answer
121
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Galois group of $X^4-k$ and $X^6-k$
I'm asked to find the Galois group of irreducible polynomials of the form $X^4-k$ and $X^6-k$ for $k\in\mathbb{Q}_{>0}$.
I think I got the solution for $f=X^4-k$ but I am not entirely sure if it is ...
- 315
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0
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38
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Galois Group of $\mathbb{Q}(3^{1/4}e^{\pi i/4},i)/\mathbb{Q}$
I'm working on this problem as part of a study guide and I ran into something that's tripping me up. The problem is, Let $K$ be the splitting field of $x^4+3$ over $\mathbb{Q}$. Find generators of the ...
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Quadratic and quartic subfields of $\mathbb{Q}(\zeta_{2023})$
Let $\zeta_{2023}=e^{2\pi i /2023}$ be a primitive $2023$th root of unity. Describe :
Quadratic subfields of $\mathbb{Q}(\zeta_{2023})$.
Quartic subfields $E$ of $\mathbb{Q}(\zeta_{2023})$ such that $...
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If there is a Galois extension with degree $[E:F] = n$, is there necessarily some subfield $K$ such that $[K:F] = m$ where $m \mid n$?
For example, consider $[E:F] = 1368$. Since $1368$ is divisible by $36$, is there necessarily some subextension $K/F$ such that $[K:F] = 36$, or is it just possible for one to exist? I have been self-...
- 1
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1
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40
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Minimum polynomial intermediate field cyclotomic extension
I'm trying to find the minimum polynomial of the generators of the subfields of the cyclotomic field $\mathbb{Q}(\xi_{13})$. We have:
$$G:=Gal(\mathbb{Q}(\xi_{13})/\mathbb{Q})=\mathbb{Z}_{13}^*$$
In ...
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