All Questions
99
questions
2
votes
1
answer
70
views
If $f(x)\in \mathbb{Z}[x]$ is irreducible (over $\mathbb{Q}$), is it always possible to find $a$ and $b$ in $\mathbb{Q}$ with $f(ax+b)$ Eisenstein? [duplicate]
My initial thought is no, simply because it seems too easy if it is true.
The simplest example of a nontrivial irreducible polynomial I could think of was $f(x)=x^2+1$. Unfortunately, $f(x+1)$ is ...
- 520
0
votes
0
answers
40
views
Artin's theorem exercise - proving that the fixed field is generated by the coefficients of the minimal polynomial
Suppose $L/K$ is a finite extension. $G$ is a finite group of $K$-automorphisms of $L$. Denote by $L^G$ the field elements of $L$ fixed by action of $G$. For any $\alpha \in L$ we write $f(t, \alpha) =...
- 474
0
votes
0
answers
129
views
Reducibility of $f \in \mathbb{Q}(\alpha)[x]$ for $[\mathbb{Q}(\alpha):\mathbb{Q}] \geq 2$ and $f \in \mathbb{Q}[x]$ Irreducible?
I'm struggling to make any progress on the following problem.
Let $K = \mathbb{Q}(\alpha)$ be an algebraic extension of $\mathbb{Q}$ with $[\mathbb{Q}(\alpha):\mathbb{Q}]=m$, and suppose that $f \in \...
- 1,399
5
votes
0
answers
99
views
Determine the galois group of this 6 degree polynomial over $\mathbb{Q}$
Determine the Galois group of $f(x)$ over $\mathbb{Q}$
$f(x)=x^6+22 x^5-9 x^4+12 x^3-37 x^2-29 x-15$
This question comes from Johns Hopkins University Fall 2018 algebra qualifying.
I have found it is ...
- 399
0
votes
1
answer
134
views
$x^n-a$ is irreducible in $\mathbb{Z}[x]$ if $a$ is not a $p$th power for $p\vert n$. [duplicate]
One can easily show that if $n$ is $2$ or $3$, and $a\in\mathbb{Z}$ such that $a$ is not an $n$th power, then $x^n-a$ is irreducible over $\mathbb{Z}$ since it has no roots. However, I believe more ...
- 598
2
votes
2
answers
164
views
Irreducibility and Galois group of the $2$-periodic points of a (cubic) polynomial
This question arose in observations made while trying to answer Intersecting cubic equations and roots.. Consider a (depressed) cubic polynomial $P(x) = x^3 + ax + b$, either over $\mathbb{Q}$ with $a,...
- 12.3k
2
votes
2
answers
125
views
Can all the roots of $ax^5+bx^2+c=0$, with real coefficients and $a,c\neq0$, be real numbers?
Let $ax^5+bx^2+c=0$ and $a,c\neq 0$ and $a,b,c$ are real numbers.
Can all the roots of this quintic equation be real numbers?
I divided each side by $a$ and I got
$$x^5+\frac bax^2+\frac ca=0$$
Using ...
- 652
1
vote
0
answers
80
views
What is the highest value of $n$ be to reduce a general polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0=0$ to $y^n+py+q=0 ?$
We know that the following facts.
The quadratic equation $ax^2+bx+c=0$ can be reduce to $y^2+p=0$
The cubic equation $ax^3+bx^2+cx+d=0$ can be reduce to $y^3+py+q=0$
The quartic equation $ax^4+bx^3+...
- 1,659
1
vote
1
answer
102
views
Is every polynomial of even degree reducible after some field extensions of degree 2?
Given an irreducible polynomial $p \in \mathbb{Q}[x]$ we're interested in how it factors after repeated simple field extension of degree $2$. So we generate a chain of fields $F_{n+1} / F_{n}$, where $...
- 11
6
votes
2
answers
401
views
Find the real root of the almost symmetric polynomial $x^7+7x^5+14x^3+7x-1$
Find the real root of following almost symmetric polynomial by radicals $$p(x)=x^7+7x^5+14x^3+7x-1$$
Here are my attempts.
The coefficients of $p(x)$ are : $1,7,14,7,-1$.
I wanted to try possible ...
- 652
1
vote
1
answer
62
views
How to obtain $\operatorname{Gal}(f\mid \mathbb{Q}_3)=A_3$ or $S_3$?
I'm doing a lot of Galois Theory lately. Now I'm more and more into $p$-adic fields. My goal is to choose an irreducible polynomial of degree $3$ over $\mathbb{Q}_2$ and $\mathbb{Q}_3$ such that you ...
- 41
4
votes
4
answers
651
views
If $f(x)$ is irreducible, is $f(x^k)$ irreducible?
Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer?
...
- 3,047
2
votes
1
answer
202
views
Finding a quartic polynomial whose resolvent cubic is given
Recently I was assigned a homework problem to determine the possible Galois groups of an irreducible quartic polynomial over $\mathbb{Q}$ that has exactly two real roots (equivalently has negative ...
- 111
2
votes
0
answers
54
views
Isomorphism of Galois Groups
$E$ is the splitting field for $f(x)=x^3-2$ and $K := E(\sqrt 5)$. We want to show that $G=\text{Gal}(K/\mathbb{Q}) \cong \mathbb{Z_2} \times S_3$.
To do this, we know that $K$ has subfields $E$ and $...
- 99
1
vote
0
answers
98
views
A polynomial is irreducible in a splitting field of another polynomial
Suppose that $E$ is a splitting field over $\mathbb{Q}$ for $f(x)=x^3-2$. We want to show using Galois theory that $g(x)=x^2-5$ is irreducible in $E[x]$.
Here is what I have so far. Assume for ...
- 115