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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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It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.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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The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$
Given two positive integers $n,m$, which positive integers $d$ appear as the degree of $\mathbb{Q}(a,b)$ for two algebraic numbers $a$ and $b$ of degrees $n$ resp. $m$?
Two necessary conditions are $\...
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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 $\...
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Extending commuting automorphisms of a field
I have a field $L$ and two field automorphisms $f, g: L \to L$ such that $f \circ g = g \circ f$. Under what conditions can they be extended to automorphisms of an extension $K \mid L$ that still ...
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Copies of topological fundamental groups inside etale fundamental groups given by different embeddings of your field into $\mathbb{C}$
Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.
Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract ...
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Limit of trace maps in finite fields
If $\mathbb{F}_{q^n}$ is a finite field with $q^n$ elements ($q$ being a power of a prime $p$) we have the trace map $tr^n_m:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_{q^m}$ such that $x\mapsto x+F^m(x)+....
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Relation - Anabelian geometry and Tate conjecture
A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.
I would like to know what is the relation between Anabelian algebraic geometry and Tate ...
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How are motives related to anabelian geometry and Galois-Teichmuller theory?
In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...
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Using Jordan's theorem to find Galois group for a polynomial
I'm trying to apply the result of Jordan's theorem (cited below) to find the Galois group for a given polynomial. My goal is to provide an example where Jordan's theorem is useful, so the polynomial I'...
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Splitting the Resultant, as when the Determinant becomes the square of the Pfaffian
The Determinant of an $n\times n$ matrix, viewed as a polynomial in the entries, is irreducible. But when it is restricted to the subspace of alternate matrices, it becomes reducible, actually the ...
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Geometric intuition for Fontaine-Wintenberger?
I asked my advisor the question in the title. He told me it was a stupid question and that I should focus on my research. Thus we're asking here.
The statement of Fontaine-Winterberger, per their ...
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Nielsen--Schreier for fields
Is it true that a subextension of a purely transcendental extension is itself purely transcendental?
In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
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How to prove that $M^G=\mathbb{F}_p[x_1\cdot v, x_1^{p\cdot(p-1)}+ v^{p-1}]$?
Let $G=Sl_2(\mathbb{F}_p)$ and $M= \mathbb{F}_p[x_1,x_2]$, where $p$ is a prime.
$M$ is a $G$-module with $(A\cdot x_1, A\cdot x_2)=(x_1,x_2)\cdot A, (\forall) A \in Sl_2(\mathbb{F}_p)$.
I have to ...
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Motivic Galois theory and Betti realizations?
Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
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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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Are polynomials with non-($S_n$ or $A_n$) Galois groups discrete?
There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $...
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Quadratic suborders of an imprimitive quartic order
Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
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