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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minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p
Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...
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Cubic polynomials with "nice" roots, which can be expressed by trig functions of rational angles
Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$.
It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...
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Cutting and pasting in Galois theory
I want to ask who was the first to use cut-paste construction in Galois theory.
This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois ...
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Is the other extreme of Hilbert Irreducibility true?
Let $K$ be a number field (or perhaps more generally a Hilbertian field). Let $X_K\rightarrow \mathbb{P}^1_K$ be a regular (i.e. without extension of scalars) $G$-Galois branched cover. Hilbert's ...
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Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
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"The Galois group of $\pi$ is $\mathbb{Z}$."
Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what sense/...
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The image of generator under an automorphism of a cyclic function field
I'm reading the proof of Lemma 4.1 [1] which says:
"Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$.
Let $Z := Gal(F/K(x))$ and we have $Z < G < Aut(F/K)$...
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A question related to Hilbert's Irreducibility Theorem
My question is whether for every extension of number fields $L\subset K$, and for every $f_0(x),...,f_n(x)$ in $K[x]$, there is some $\alpha\in L$ such that $$f_n(\alpha)T^n+...+f_1(\alpha)T+f_0(\...
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Solving the cubic by "radicals" in characteristics 2 and 3
This question has no justification other than a bit of fun.
We all know that the cubic is solvable by "radicals" ($\root2\of{}$ and $\root3\of{}$) in characteristics $\neq2,3$. The formula was ...
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Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?
Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...
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Constructive proof of algebraic elements forming a subfield
Let $F \leqslant E$ be a field extension.
If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$:
$[E(a,b):E]$ ...
user11863
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Galois theory for polynomials in several variables
I feel a bit ashamed to ask the following question here.
What is (actually, is there) Galois
theory for polynomials in
$n$-variables for $n\geq2$?
I am preparing a large audience talk on Lie ...
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What does Gal(Q_p/Q) mean? [closed]
What does
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.)
If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
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Automorphisms of local fields
It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...
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Why should the anabelian geometry conjectures be true?
I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$...
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Galois groups at closed points from Galois group at generic point?
Consider the finite map $\mathbb{A}^1_\mathbb{Q}\rightarrow \mathbb{A}^1_\mathbb{Q}$ given by $z\mapsto z^5-z$. The fiber over generic point is the field extension $\mathbb{Q}(t)[z]/(z^5-z-t)$ over $\...
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Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
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what part of using vieta's formulas violates quintic non-solvability? [closed]
You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas.
You can solve this system of nonlinear equations using Newton's method and the Jacobian. ...
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Galois theory and algorithms
Steven Weintraub's book {\em A Guide to Advanced Linear Algebra} includes the following remark:
"Of course, there is no algorithm for factoring polynomials, as we know from Galois theory."
I can't ...
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What is the general statement of Hilbert 90?
I know two generalizations of Hilbert 90, but I don't if there is a statement that contains both:
The first statement
Let $K$ be a field, then $H^1(Gal(K), GL_n(K^{sep}))=0$.
The second statement
Let $...
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Quadratic extension of quadratic extension
Let $K_1$ be a perfect field. Let $K_2/K_1$ and $K_3/K_2$ be quadratic extensions. Let $K_4/K_3$ be the Galois closure of $K_3$ over $K_1$. Is it true that either $K_3 = K_4$ or $K_4/K_3$ is quadratic ...
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Multiplication of matrices in GF(2) and R
$H$ is an $n \times n$ matrix with elements in $ \{ -1,1 \}$
$G$ is an $n \times k$ matrix with elements in $GF(2)$ and also upper triangular, invertable
$m$ is an $k \times 1$ vector with elements ...
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More on "Transalgebraic Theories" (a 19th century yoga)?
Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful '...
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Searching for polynomials with squarefree discriminant
In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually ...
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Is there an alternative formula for solving cubic equations?
It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
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$Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$
Observe that we have $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$.
More generally, assume that $K$ is a finite extension of Q. Is there any $\alpha \in K$ such that $K=Q(\alpha^n)$ for every $n \in N$?
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A family of polynomials with symmetric galois group
Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero:
$f_n(x,y)=(x+y)^n+(x-1)y^n,$
for $n\geq 3$. I can prove that $f_n(x,y)$ has an irreducible factor of ...
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Motivation for the proof of Hilbert's Theorem 90
The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots \sigma^{n-1}...
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What Dirichlet doesn't tell...
Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural ...
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Galois connections
I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the ...
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What are the different theories that the motivic fundamental group attempts to unify?
I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In http://www.math.ias.edu/files/deligne/...
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Galois groups of number fields
It seems that it is conjectured, that the absolute Galois group of a number field determines already the number field up to isomorphism.
I would like to know if there is a profinite group G such that ...
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What is the purpose of tangential base-points?
Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree $1$)...
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Expressing Galois actions on fundamental groups explicitly
Let $X$ be some variety over $\mathbb{Q}$, and let $\pi_1(X\times_{\mathbb{Q}}\mathbb{C},x)$ denote its (topological) fundamental group. As is well known $Gal(\mathbb{Q})$ acts on this fundamental ...
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In Riemann Existence, what is the interpretation of the space of complex-geometric points?
I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:
Question
Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{...
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Grothendieck's Galois theory without finiteness hypotheses
This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any ...
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On the inverse Galois problem
Q: What is the "simplest" finite group $G$ for which we don't know how to realise it as a Galois group over $\mathbf{Q}$ ?
So here the word simplest might be interpreted in a broad sense. If you ...
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Splitting a polynomial with one root
Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$?
I am mostly interested in the ...
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Extensions obtained adding torsion points of an elliptic curve
When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{...
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Effective Chebotarev density results for arbitrary number fields
So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define
the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of
the ...
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On the field of invariants of a finite group
So let $G$ be a finite group and let $\iota:G\rightarrow S_n$ be an embedding of $G$ in a symmetric group of degree $n$ for some fixed integer $n$. Let $K$ be a fixed field of characteristic $0$. The ...
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What is a "best" transcendence basis for R/Q ?
It is easy to show, using the axiom of Zorn, that there exists a transcendence basis for $\mathbb{R}/\mathbb{Q}$, i.e. a set $S$, algebraically independent over $\mathbb{Q}$, such that $\mathbb{R}/\...
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nth-powers and degree n polynomials with coefficients in field extensions
Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks
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Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
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Is the Leopoldt conjecture almost always true?
The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\...
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Is the etale fundamental group of Spec(Z)\{p_1,...,p_n} finitely presented?
(of course not, it's usually uncountable; I really mean is it the profinite completion of a finitely presented group).
By definition, $\pi_1^{\operatorname{et}}(\operatorname{Spec}(\mathbb Z)\...
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Norms in Galois extensions
Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both $\mathrm{...
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Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
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Where is a good place to start learning about the Grothendieck-Teichmuller group?
I've had a desire to get some sort of handle on the Grothendeick-Teichmuller group for years now, but I've always felt that I could never find a source that was introductory and readable.
The obvious ...
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Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
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