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6
votes
0answers
183 views

Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
-5
votes
1answer
280 views

Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...
5
votes
2answers
370 views

Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$. What is the isomorphism class of the inertia group $I_p$, ...
5
votes
0answers
162 views

Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial? We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...
8
votes
2answers
383 views

Inverse Galois problem for $GL_2$ of a compact local ring

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring) For which $A$ as above is there a ...
5
votes
0answers
204 views

On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving, $$-24a+36a^2-12a^3+a^4=z^2\tag1$$ which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...
1
vote
1answer
83 views

Separable extensions of henselian fields

Let $(k,v)$ be a henselian field, with $\mathcal{O}$ and $\bar{k}$ being respectively its valuation ring and its residue field. If $K/k$ a finite separable field extension (on which $v$ thus extends ...
2
votes
2answers
408 views

Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

This is a re-post from MSE (because I did not get the kind of answer I wanted even after offering a bounty). At the outset I must mention that I don't have a fairly working knowledge of Galois Theory ...
3
votes
1answer
168 views

Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$

This is related to a question on Math Stack Exchange. Given a rational number $a>0$ and an $n\in\mathbb{N}$ such that $x^n - a$ is irreducible over $\mathbb{Q}$, it is known that every subfield of ...
0
votes
1answer
186 views

Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation $a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...
2
votes
0answers
176 views

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? ...
6
votes
0answers
380 views

Algebraic Closure of the field of rational functions

Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is ...
5
votes
1answer
155 views

Coverings/Cech cohomology of totally disconnected spaces

For any topological space $X$ we have a natural functor $\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$ from the category of coverings of $X$ to the category of functors $\pi_1(X) ...
13
votes
1answer
549 views

When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
3
votes
0answers
187 views

How to handle a polynomial whose roots exhibit obvious symmetry

I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one All others have their roots arranged in a similar ...
32
votes
2answers
3k views

Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
5
votes
1answer
219 views

Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...
5
votes
4answers
801 views

Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false? We know a lot of things that would be true if the Riemann Hypothesis holds. ...
0
votes
1answer
227 views

Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3} - 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E + 135\, {\beta}_{\pm} ...
3
votes
0answers
173 views

Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
14
votes
0answers
309 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
2
votes
1answer
176 views

What are some good references on the Galois theory, factorization, or minimality of differential equations?

Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u ...
3
votes
1answer
123 views

Density of tuples of conjugate algebraic numbers

One can see that algebraic numbers are dense in the complex plane by just looking at quadratic polynomials. I am interested in a "higher order" density of algebraic numbers. More specifically: is it ...
10
votes
1answer
239 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
9
votes
2answers
454 views

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 ...
24
votes
0answers
699 views

On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...
2
votes
0answers
178 views

Explicit formula for $\sin\frac{\alpha}{3}$ [closed]

Recently I was thinking on the following problem: find all $\alpha\in R$ such that $\sin\frac{\alpha}{3}$ has an algebraic expression involving only rational powers over rational constants and ...
4
votes
0answers
175 views

On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$

I. Given the roots $x_i$ of the general cubic, $$x^3+c_2x^2+c_1x+c_0=0\tag1$$ with $c_i \in \mathbb Q$, it is easy to show that the expression, $$F_3 = (x_1^{1/3}+x_2^{1/3}+x_3^{1/3})^3$$ is an ...
11
votes
0answers
322 views

Evaluating products of cyclotomic polynomials at roots of unity

Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
6
votes
1answer
335 views

abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of $$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$ obtained from the homotopy exact sequence of the étale ...
3
votes
1answer
120 views

Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by, $$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$ with discriminant $D = (7 + ...
0
votes
0answers
116 views

construct totally real cubic fields

Given some $e_i=0$ or $1$ for $1\le i \le 3$. I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where ...
9
votes
1answer
277 views

Variance of the roots of a complex polynomial

Let $P\in\mathbb{C}[X]$ be a complex polynomial of degree $n\geq 2$ with complex roots $\alpha_1, \alpha_2,\ldots, \alpha_n$. My question is about the existence of a formula for the variance of the ...
9
votes
0answers
347 views

Do we know that 'most' finite groups are Galois groups of number fields?

The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...
6
votes
2answers
586 views

Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether ( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...
5
votes
1answer
462 views

Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
14
votes
2answers
1k views

Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all ...
2
votes
0answers
59 views

Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, ...
2
votes
0answers
153 views

What are the minimal degrees of the real and imaginary part of an algebraic complex number? [closed]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...
7
votes
0answers
370 views

The construction of the 257gon

If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...
48
votes
2answers
1k views

Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?
3
votes
2answers
188 views

efficiently checking that a field extension is Galois

Let $K \subset L$ be an algebraic extension of fields finitely presented over a prime field or over an algebraically closed field. Is there an efficient procedure to check that $L/K$ is Galois? To ...
7
votes
2answers
636 views

Galois groups and prescribed ramification

What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
13
votes
2answers
944 views

What justification can you give for the fact that “most ODEs do not have an explicit solution”?

What justification can you give for the fact that "most ODEs do not have an explicit solution"?
8
votes
2answers
724 views

Is a Galois extension of the rationals determined by its set of completely split primes?

apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the ...
11
votes
4answers
2k views

Can a sum of roots of unity be an integer?

Let $n \geq 2$, $H \lneq (\mathbb{Z}/n\mathbb{Z})^*$, $\zeta_k$ a primitive $k$-th root of unity. Is it possible that $$\sum_{h \in H} \zeta_k^{h} \in \mathbb{Z}$$ for every $k$ dividing $n$ such that ...
4
votes
1answer
367 views

q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity. Prove or disprove that the ...
14
votes
2answers
2k views

Is there an algebraic number that cannot be expressed using only elementary functions?

(this is basically a repost of a question I asked at M.SE last year) Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that ...
6
votes
3answers
400 views

Finite extension of local fields

Can a (higher) local field have uncountably many finite (seperable) extensions?
6
votes
1answer
233 views

Comparison of finite field extensions of $\mathbb{C}(t)$

Let K be a finite field extension of $\mathbb{C}(t)$. Then $K$ is isomorphic to the field of meromorphic functions on a compact Riemann surface $X$ with genius $g$. By an argument similar to the proof ...