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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Rank of sum of Galois conjugates of a matrix
Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in \mathrm{...
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Which polynomials arise as formulas for a conjugate
For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a ...
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Is H^2(W_p,C^times) well-known?
Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated?
(This group ...
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An advanced exposition of Galois theory
My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would ...
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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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Cyclotomic extensions with split Galois group
$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$
Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that
$$
\Gal(E/\Q) \simeq\Gal(E/\...
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How many algebraic integers exist with degree $\leq k$ and bounds on the modulus of all Galois conjugates?
The precise question is the following:
Question: Can one reasonably bound the number of algebraic integers $\alpha$ of degree at most $k$ - that means there exists a monic integer polynomial $p$ ...
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Computing only the order of Galois group (not the group itself).
My question is related to this one: Computing the Galois group of a polynomial.
I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself.
...
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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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What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
I think the answers for the first few degrees ($n$) are:
$n=2$, $S_2$
$n=3$, $S_3,A_3$
$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)
$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ (...
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Are all solvable groups *regularly* realizable over Q(x)?
It is known for Hilbertian fields that all groups that are abelian, solvable, $A_n$ or $S_n$ are realizable over them. $\mathbb{Q}(x)$ is one such field, but it's not obvious that the extensions that ...
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Is it possible to recover the degree of a field extension from a list of elements and the ground field?
I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
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Congruences mod primes in Galois extensions
I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields
$$\mathbb{Q}\...
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When are the intermediate fields totally ordered?
The groups whose subgroups are totally ordered by inclusion are easy to classify; they are subgroups of $\mathbb{Z}/p^{\infty} = \text{colim } \mathbb{Z}/p^k$ for some prime $p$, and thus $\mathbb{Z}/...
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Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?
Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...
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Fields with trivial automorphism group
Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know.
Every prime field has trivial automorphism group.
Suppose L is a separable finite extension ...
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Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
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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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Proof of the result that the Galois group of a specialization is a subgroup of the original group?
I have been using the following result:
Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same degree,...
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History of the Normal Basis Theorem
The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space ...
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Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups
Coming up with examples of $D_8$-covers of $\mathbb{C}(x)$ is easy. For example:
$Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$
defines a $...
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How do I visualize finite covers of curves over non-algebraically closed fields?
If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
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Normal subgroups of the Galois group
I am trying to teach myself Galois theory. Is it true that every for a field extension K->L, that every normal subgroup of Gal(L:K) is of the form Gal(L:M) for some intermediate field M, ie K->M->L?
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Galois theory of endomorphism rings of irreducible representations
Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring.
What is known about ...
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Computing stable reduction of finite covers of curves in practice
The general theory is described in various places, but I'll be following (sketchily) the description of this process appearing in section 1 of Bouw and Wewers' "Reduction of covers and Hurwitz spaces"....
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Symmetric polynomials theorem
Hello all, I would appreciate comments on the following question:
A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = k[s_1,...,s_n]$,...
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algebraic numbers of degree 3 and 6, whose sum has degree 12
This question is related to Degree of sum of algebraic numbers. Forgive me if
this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the ...
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Galois Group as a Sheaf
I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
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Probability of an extension being normal
Let $P$ be the probability that an elliptic curve with a rational point has an infinite number of rational points. From what I understand, the value of P is unknown.
This got me thinking about a ...
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Primitive element theorem without building field extensions
Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$...
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Why do generic polynomials work in reality?
I understand that a generic $G$-polynomial $f(t_1,...,t_n)[X]$ over field $k$ has Galois group $G$ over $k(t_1,...,t_n)$. And basically any $G$ extension of $k$ should be generated by a realization of ...
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Galois theory timeline (II)
This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
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Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
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Is the (regular) inverse Galois problem known for the field C(x,y)?
I'd be surprised if somebody proved the inverse Galois problem for $\mathbb{C}(x,y)$, but I wanted to make sure.
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Maximal extension almost everywhere unramified and totally split at one place
Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ ...
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what is the first Galois cohomology group of the Galois module End(T_l(A)) for some abelian variety A over a finite field k and l some prime number different from the characteristic of the base field?
According to Serre's book 'Galois cohomology', Galois chomology group are always torsion, but it seems to me that H^1(k, End_{Z_l}(T_l(A)))=coker(Frob-1) on End_{Z_l}(T_l(A)), which has the same Z_l ...
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A split short exact sequence of algebraic fundamental groups
If we have a variety, $X$, over a field, $k$, and $x$ is a geometric point of $X$, and let $\bar x$ be a geometric point of $X_{k^s} := X \times_k k^s$ above $x$ then we have the following short exact ...
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method of finding roots of polynominal equations with arithmetic operations and roots and other functions
Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on ...
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
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What is the Galois group of a polynomial over a finite field? [closed]
If I have a polynomial which I've factorised into irreducibles over GF(p), p prime, and it doesn't have any repeated factors, then what is its Galois group over this finite field (and what is the ...
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A Galois Theory Computation
Excuse me for the specificity of this question, but this is a silly computation that's been giving me trouble for some time.
I want to explicitly realize the order 21 Frobenius group over ℂ(x), ...
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Surprising Analogue of Q
I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer.
Manish Kumar proved that the commutator subgroup ...
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Galois group of a product of irreducible polynomials
Hello
Suppose given a polynomial $P=Q_1\cdots Q_k$ of degree $n$, where each $Q_i$ is irreducible. Suppose also that I know the Galois group $G_i$ (over the rationals) of each irreducible factor $Q_i$...
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A special integral polynomial
Given $n \in \mathbf{N}$,is always possible to construct a monic polynomial in $\mathbf{Z}[x]$ of degree $2n$, whose roots are in $\mathbf{C} \setminus \mathbf{R}$ and whose Galois group over $\mathbf{...
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How to show the galois group of a polynomial is not an alternating group?
I am trying to prove that a certain class of polynomials have symmetric galois group.
Using the Newton polygon, I have shown that the galois groups of these polynomials are transitive on $k$-sets for ...
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For a given finite group G, is there a cover of P^1 over Q s.t. over C it's G-Galois?
For any finite group, G, we can find a cover of ℙ1ℂ which is G-Galois. The regular inverse Galois problem is equivalent to there existing such a cover that descends with action to ℚ. ...
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Number theory textbook based on the absolute Galois group?
I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting ...
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What, precisely, is the relationship between "fields of moduli" and "moduli spaces"?
Notation
The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ1 be a G-Galois cover, where everything is over the algebraic closure of ...
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Field of Definition of a Meromorphic Function
Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field,...
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Explicit expression of an alternating polynomial in characteristic $2$?
Although the question is easy to pose, I think some background will help to motivate it, so I'll start with it.
Consider variables $X=(X_1, \ldots, X_n)$ over a field $K$ and the elementary symmetric ...
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