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.
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Relationships among constructions of fundamental group for schemes
There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
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Interpolation of families of local fields
Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
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“Algebraization" of $p$-adic fields
Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
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Having a separable extension of degree $n$ implies having a Galois extension of degree $n$?
I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of ...
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Cyclic cubic extensions and Kummer theory
The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field ...
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Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$
I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
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Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate?
Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$.
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Does there exist a number field, unramified over a predetermined finite set of primes of Q, such that the inverse regular Galois problem is correct for that number field?
The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...
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Recognizing the Galois group from the field discriminant
Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
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Shafarevich's theorem about solvable groups as Galois groups
I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
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The inverse Galois problem, what is it good for?
Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience ...
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Normal closure and separable elements
Let $K\subset E\subset\bar{K}$ be field extensions, $\bar{K}$ an algebraic closure of $K$. Denote $E_s$ the field of separable elements of $E$ over $K$, denote $\tilde{E}\subset\bar{K}$ the normal ...
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Unique factorisation of prime geodesics?
In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
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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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Centraliser of an absolute Galois group
Let $K$ be a finite extension of $\mathbb{Q}_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ?
If yes, how can I ...
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What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
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Quintic polynomials generating cyclic extensions
We know that a cubic equation generates a cubic cyclic extension iff it has a perfect square discriminant. Now I am wondering if there is a similar condition for quintic polynomials. So I am trying to ...
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Characters of the kernel of the norm map of an extension of local fields
Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm ...
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What is the dimension of the mathematical universe?
Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...
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Grothendieck's "La longue Marche à travers la théorie de Galois"
It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.
Is there any way to obtain a copy (online or not) of "La ...
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Sign preserving Galois automorphisms
I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...
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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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Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?
Let $K$ be a $p$-adic local field, for example $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true ...
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Finding a cyclic cubic extension of a field
Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
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Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
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Galois groups of a family of equations
I investigated a family of polynomials given by for $n>1$:
$x^n+(x+1)^n+...+(x+n-1)^n=(x+n)^n$
And I found, with the help of Magma, that the Galois group is $S_n$ for $n$ up to a hundred. The ...
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Example of an algebraic number of degree 4 that is not constructible
The number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$ with
$a:=\frac{\sqrt[3]{18+2\cdot\sqrt{65}}}{2}+\frac{2}{\sqrt[3]{18+2\cdot\sqrt{65}}}$ is a root of the irreducible polynomial $x^4-6x+...
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Inverse Galois problem for $2$-groups with an involution as complex conjugation
It is known that the inverse Galois problem for solvable groups was solved by Shafarevich. My question is the following: given $G$ a finite $2$-group and $s$ an element of order $2$ in $G$. Can we ...
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Quadratic equation over a global field of characteristic 2
Let $F=\mathbb F_{2^n}(t)$, and let $f=x^2+ax+b\in F[x]$. Is there any necessary and sufficient condition for $f$, depending on its coefficients, to have a root in $F$? I'm not interested in finding ...
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Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?
If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...
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Field of constants of a Galois extension of function fields
Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a ...
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Geometric fundamental group and algebraically closed residue field
my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
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Bounding the degree of an algebraic extension containing solutions to polynomials
Also posted on math.stackexchange...
Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by ...
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Gauge equivalence between operators
I have tried to figure out the following problem for some time now, but with little success:
Let $ \mathcal{L} $ be a third order linear differential operator with coefficients in $ \mathbb{C}(X) $. ...
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Abelian group extensions
Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...
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When is a bilinear form equivalent to a trace form?
Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...
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Realization of the primitive action of a wreath product in a Galois group
Let $f$ be a polynomial over a field $K$ of degree $n$ such that $f(x^2)$ is separable. Assume that the Galois group $G$ of (a splitting field of ) $f(x^2)$ is maximal, that is to say, the wreath ...
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Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves
I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations.
Let $E$ be an elliptic curve over $K$, ...
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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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Interpretation or application of this analog of minimal polynomial
Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I ...
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Galois theory timeline
A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...
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Symmetrizing with respect to Galois Group: Trace and Norm
In invariant theory the Reynold's Operator gives rise to an element invariant for that group.
For a Galois extension $K/F$ with $K=F[\alpha]$ the trace of $\alpha$ is an element of $K$. If $\alpha$ ...
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Elements of finite fields with many powers of trace zero
Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\...
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Galois Transcendental Field Extension has characteristic Zero
Let consider a Galois transcendental field extension $T/K$, therefore for each subextension $L$ of $T/K$ we have $T^{\operatorname{Aut}(T/L)} = L$.
My question is how to prove that this conditions ...
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Why is $K_{\upsilon}|K$ separable for a global field $K$?
I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...
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Family of irreducible trinomials over finite fields [closed]
Is there any "famous" family of trinomials over finite fields?
For example over $F_2$ we have
$$
f(x)=x^{2\times 3^k}+x^k+1
$$
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Galois theory, topos vs fundamental groups
Classical Galois theory states that the etale topos X of a field k is equivalent to the classifying topos of the absolute Galois group of k.
(Marc Hoyois, Higher Galois theory, $\S$3, arXiv:1506....
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Proof of Witt's result about quaternion extensions
I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{...
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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 ...
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Minimal polynomial of a trigonometric number
I am trying to calculate the minimal polynomials of $h_{1}=-\cos(\pi/n)-\sqrt{\cos(2\pi/n)}$ and $h_{2}=-\cos(\pi/n)+\sqrt{\cos(2\pi/n)}$ when $n$ is odd. I think (and numerical calculations suggest ...
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