14
votes
1answer
716 views

Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$? The standard irreducibility criteria seem to fail.
7
votes
1answer
236 views

Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$

Theorem. The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root. Question. What ...
2
votes
2answers
126 views

Sextic resolvent has no rational root

An irreducible quintic $f(x)\in\mathbb{Q}[x]$, is solvable by radicals if and only if its sextic resolvent $\theta_f (y)=(y^3+py^2+qy+r)^2-2^{10}\Delta(f)y$ has a rational root ($\Delta$ is the ...
3
votes
2answers
162 views

“Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ? Is there some kind ...
17
votes
1answer
974 views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
9
votes
3answers
1k views

An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index. There is an approach that ...
3
votes
2answers
336 views

The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...
0
votes
0answers
106 views

On the Brioschi-like quintic $v^5 - 5d v^3 + 10 d^2 v - d^2 =0 $

The general quintic can be transformed in radicals using a rational Tschirnhausen transformation to the one-parameter Brioschi quintic, $$u^5 - 10c u^3 + 45 c^2 u - c^2 = 0\tag{1}$$ which can be ...
1
vote
1answer
135 views

Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
10
votes
1answer
382 views

Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...
0
votes
2answers
162 views

power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...
3
votes
1answer
478 views

When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?! In another words, what could you impose on the coefficients to ...
2
votes
1answer
231 views

A cubic polynomial which contains a linear factor with irreducible residual quadratic form

Let $f(x)\in\mathbb{Z}[x_{1},\dots,x_{n}]$ be a cubic homogeneous polynomial, which factors as $f(x)=g(x)h(x)$ over $\mathbb{C}$ with $\mathrm{deg}(g)=1$ and $h$ irreducible over $\mathbb{C}$. Assume ...
2
votes
1answer
526 views

Solving polynomial equations in radicals provided all roots are rational

This question is related to this question of Joseph O'Rourke and this question of mine. Question. Let $f$ be a polynomial with integer coefficients. Suppose that all roots of $f$ are rational. ...
6
votes
1answer
574 views

Can roots of any polynomial be expressed using Eulerian function?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it is claimed that roots of any ...
4
votes
1answer
830 views

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 ...
-4
votes
2answers
963 views

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. ...
7
votes
1answer
1k views

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. ...
4
votes
2answers
1k views

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 ...
6
votes
1answer
395 views

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 ...
12
votes
2answers
1k views

Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...
6
votes
0answers
199 views

Involution on sextic polynomials?

The strangeness of $Aut(S_{6})$ suggests the following question: Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be the zeros of a 6th degree polynomial $P$ over a field $F$. Let ...
9
votes
2answers
422 views

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 ...
2
votes
2answers
703 views

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 ...
42
votes
4answers
3k views

Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer. Let $a$ and $b$ be algebraic numbers, with respective degrees ...
2
votes
0answers
1k views

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 ...
8
votes
1answer
645 views

When the splitting fields of shifted generic polynomials are linearly disjoint?

Let me start by rigorously pose my question. Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + \cdots + T_n$, be the ...
5
votes
2answers
876 views

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 ...
9
votes
2answers
1k views

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 ...