Questions tagged [galois-theory]
Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.
263 questions with no upvoted or accepted answers
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Finite groups $G$ such that every Galois extension of group $G$ is totally real
This is an extended repost of the following question asked on MSE two weeks ago.
I was playing with Galois extensions of small degree while preparing the final exam for my course in Galois theory, ...
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312
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Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
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357
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The Galois resolvent in Lagrange
In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...
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221
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Field whose absolute Galois group is $\mathbb{Z}_p$
Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups.
Is there a ...
- 53
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135
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Permutation group with a nice lattice of block systems
Let $X$ be a finite set and $G$ be a transitive subgroup of the symmetric group on $X.$ Recall that a (complete) block system for this action is a partition of $X = B_1 \cup \cdots \cup B_k$ into ...
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549
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Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
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146
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Local Class field theory for curves over $p$-adic fields
Let $X$ be a smooth curve over $\operatorname{Spec}\mathbb{Q}_p$ and $P\in X(\mathbb{Q}_p)$. Let $K_P\simeq \mathbb{Q}_p((T))$ denote the completion of the function field of $X$ at $P$. What is known ...
user130124
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139
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Galois elements determined by action on $n$-th roots of rationals?
Can there be an element $\sigma$ of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, other than the identity and complex conjugation, which is completely determined up to conjugacy by its action on $\sqrt[n]r$...
- 21
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Classification of mod p Galois Representations for l not equal to p
Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
user130124
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Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character
Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...
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59
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Classes of curves with "determinant-like operation"
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
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Does the $G$-norm coincide with the ordinary norm for "quasi-$G$-Galois" extensions
Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...
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Lie algebra of derivations for a transcendental field extension and intersection fields
Suppose that $L$ is a finite Galois extension of the field $K$.
If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$
where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...
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Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
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residue fields of smooth $\mathbf{Q}$-algebras
Let $A$ be a $\mathbf{Q}$-algebra. We say $A$ is "residually abelian", if there exists a maximal ideal $\mathfrak{m}$ of $A$ whose residue field $\kappa(\mathfrak{m})$ is a Kummer extension of an ...
user92332
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Splitting fields and algebraic closure for generalized polynomials
I'm looking for a reference on splitting fields for 'generalized polynomials' over non-Archimedean fields with exponents in a non-Archimedean discretely ordered value class.
To be precise, let $\...
- 10.1k
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Problem with a proof of Wilson's 'Profinite groups'
(Crossposted on StackExchange Mathematics: https://math.stackexchange.com/questions/2391626/problem-with-a-proof-of-wilsons-profinite-groups)
I need help with the proof of Proposition (3.1.3) given ...
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Existence of "splitting objects" and algebraic closures
Definition 0. Let $(\mathbf{C},U)$ denote a concrete category. Let "splits" be a desirable property that pointed $\mathbf{C}$-objects may or may not satisfy. Let $(X,x)$ be a pointed $\mathbf{C}$-...
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complex numbers over algebraic numbers, continuous cohomology
This question is related to this one. Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from the algebraic closure of the field of rational numbers to the field of complex numbers. Is ...
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Neat applications of Galois descent?
I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space ...
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Characterization of torsors which are locally trivial in terms of descent
Let $\mathsf C$ be a category and $G$ an internal group. Suppose $\mathsf C$ is finitely complete, so that $\pi_2:G\times B\to B$ is an internal group in $\mathsf C_{/B}$ for every $B$. A $G$-bundle ...
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107
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Complex Lie inverse Galois problem
My question is about the inverse Galois problem for infinite dimensional complex manifolds. If $K$ is the field of meromorphic functions over a complex manifold $M$ and $G$ is a finite or infinite ...
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203
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What's the connection between Galois objects and Galois closed objects?
An object in a free coproduct completion is Galois closed if it has no nontrivial coverings, i.e every covering morphism is split by the identity.
An object of a Galois category is a Galois object if ...
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166
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Relative Leopoldt defect
Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...
- 1,066
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90
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Conditions to check whether a prime ramifies in a certain way
Let $p$ be a prime number and let $f = a_n x^n + \cdots + a_0$ be an irreducible polynomial of degree $n \geq 3$ with a root $\alpha$. We say that $p$ ramifies in a powerful way over $K = \mathbb{Q}(\...
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Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$
Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers.
Question 1: Is it true that $\mathbf{C}$ is ...
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Which fields have no extensions of degree divisble by a fixed prime?
Let $p$ be a prime. What are the most general examples of a field $K$ such that for any finite extension $L/K$ the degree $[L:K]$ is prime to $p$?
Certainly, there are algebraically closed examples ...
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Reference request: Cohomology of Elliptic Curves
Is it true that the group
$$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$
is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility?
...
- 1,805
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150
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Galois correspondence for action of general linear group on purely transcendental extension
For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a ...
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lift isomorphic in a sufficiently thick fiber
Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$.
We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.
Let $d=val(\Delta(P))$ where $\...
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About Schanuel's conjecture
I just read an article of Ram Murty about transcendence of special values of L-functions, and it seems that Schanuel's conjecture plays a crucial role in it. So given a positive integer $n$, let's ...
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On Artin-Hironaka lemma and Galois theory
Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point.
Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...
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Explicit defining equations for the Leopoldt locus
Let $F$ be a number field, which we assume for simplicity to be Galois and totally real. Set $\mathcal{O}_p=\mathcal{O}_F\otimes_{\mathbf{Z}}\mathbf{Z}_p$. The norm map on $\mathcal{O}_F$ extends ...
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Decomposition group vs Galois group of completed extension for height > 1 primes
Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...
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What do you call an algebraic element with the property that the generated field extension is normal?
Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...
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Splitting of prime ideals in non-Dedekind domains?
This is a follow-up to this question. So that you don't have to flick back and forwards I'll briefly summarize:
My original question was on how to prove that a polynomial $g(x)$ obtained from $f(t,x)...
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
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inverse Galois problem on cyclic groups
It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
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Kernel of inflation of $H^2$ in Galois cohomology
For a finite set $T$ of primes let $G_T$ denote the Galois group of the maximal extension $K_T$ of $\mathbb{Q}$ which is unramified outside $T$.
Let $S$ denote a finite set of primes and let $S' = S\...
- 2,907
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Is the Galois closure of a $p$-adic Lie group extension also a $p$-adic Lie group extension?
Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
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Inverse Galois problem for G with a derived sequence of length 2
For a finite group G with a derived sequence of length 2, please tell me how to specifically construct a field that is a Galois extension of $\mathbb{Q}$ and whose Galois group is G.
I made some edits ...
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When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 341
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Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
- 14.2k
1
vote
0
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103
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Abelian extensions of the rationals
Let $E$ and $F$ be finite abelian extensions of $\mathbb{Q}$ such that $E\cap F=\mathbb{Q}$. (You could take $E$ and $F$ as cyclotomic fields if that makes my question easier.) Set $K:=EF$ (the field ...
- 385
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Galois connection for homeomorphisms
let $M = \mathbb{R^2}$ and $X = \{0\}$ and $G = Aut_X(M)$ the group of homeorphisms fixing $X$ (pointwise). Then we have, in analogy to classical Galois theory for field extensions, a Galois ...
- 11
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61
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Galois action on blow-ups related to field extensions of infinite degree
Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
- 4,547
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124
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$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
- 763
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175
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Algebraic numbers with a polynomial property
In my research I faced with an intricate construction of an algebraic number with some properties.
Problem. For which classes of polynomials $P(X,Y)\in \mathbb{Z}[X,Y]$, we have the following property....
- 515
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0
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135
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Reference request: arithmetical implications of an ambient Galois extension
This is a cross-post of a question I asked on StackExchange. See there for further details.
Let $L/K$ be a Galois extension of algebraic number fields of finite degree over $\mathbb{Q}$, with group $G$...
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