All Questions
214
questions
0
votes
1
answer
27
views
Determining the Equality of Two Field Extensions
Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
- 730
2
votes
1
answer
83
views
If $F/K$ is normal extension and $f \in K[x]$ irreducible and $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ in $F[x]$ then $m_{i}=m_{j}$ for all $i,j$
So I have that question:
Let $F/K$ be a normal extension and $f$ irreducible polynomial in $K[x]$ assume that $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ where $g_i$ is irreducible in $F[x]$ ($m_i \geq 1$) then ...
- 115
2
votes
1
answer
127
views
Irreducibility of a Polynomial with Prime Exponents
Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$
where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
- 637
0
votes
0
answers
69
views
Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$
Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
- 2,066
0
votes
0
answers
46
views
Reducing an inseparable polynomial over the same field to a separable polynomial over a field
Description: Let $F$ be a perfect field and $p(x)$ a polynomial over $F$ with multiple roots. Show that there is a polynomial $q(x)$ over $F$ whose distinct roots are the same as the distinct roots of ...
5
votes
1
answer
113
views
Irreducible $p(x) \in \mathbb{Q}[X]$ with roots $r, s$ such that $rs = 1$.
If $p(x) \in \mathbb{Q}[X]$ is irreducible and has two roots $r,s$ such that $rs = 1$, then $p(x)$ is of even degree.
I'm not sure how to solve this problem. My initial idea was to consider the field ...
- 464
0
votes
2
answers
83
views
Polynomial factorisation over quotient ring
Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x] $
Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$.
I already showed that f is ...
- 93
2
votes
1
answer
155
views
$f(X) = X^9 - 2dX^6 + 3d^2 X^3 - d^3$ irreducible in $\mathbb{Q}[X]$ for given cube-free integer $d>1$.
Here's what I've tried:
I've checked that there's exactly one real root for this polynomial (that's easy to prove i.e. there's at least one real root since it has odd degree and it is easy to show ...
2
votes
1
answer
91
views
Show $x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Q}(\sqrt[5]{2})$
I have proved that $f(x)=x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Z}_2$, so it's irreducible in $\mathbb{Z}$, hence in $\mathbb{Q}$.
I know $f(x)=x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Q}(\...
- 311
3
votes
2
answers
127
views
Is $x^6 + bx^3 + b^2$ irreducible?
Let $b\in \mathbb{Q}^*$ be rational number. We factorise $x^9-b^3\in \mathbb{Q}[x]$ and obtain $$x^9-b^3=(x^3-b)(x^6+bx^3+b^2).$$
Is the polynomial $x^6+bx^3+b^2$ irreducible?
If $b=1$ we get a ...
- 3,979
6
votes
1
answer
105
views
Does $\sqrt a + \sqrt b$ have a four way conjugate?
Let $a, b$ be rational numbers that are not perfect squares. Consider the set $S = \{\sqrt a + \sqrt b, \sqrt a - \sqrt b, - \sqrt a + \sqrt b, -\sqrt a - \sqrt b\}$.
If $p$ is a polynomial with ...
- 4,450
1
vote
2
answers
69
views
Factorization over $\mathbb{Q}$ and $\mathbb{Z_{41}}$
Factor $f(x) = x^4+1$ over $\mathbb{Q}$ and over $\mathbb{Z_{41}}$.
1)I can't factor $f(x)$ over $\mathbb{Q}$ because $f(x+1)$ is irreducible by Eisenstein's criterion.
2)I don't know where to start:
...
- 45
0
votes
2
answers
50
views
$x^{\frac{p-1}{2}}+1$ is reducible in $\Bbb Z_p[x]$ [closed]
Let $p$ be an odd prime. Prove that the polynomial $f(x) = x^{\frac{p-1}{2}}+1$ is reducible in $\Bbb Z_p[x]$ and factor $f(x)$ into irreducible polynomials in $\Bbb Z_p[x]$.
I've been struggling ...
- 103
1
vote
0
answers
77
views
Ask for help on proving irreducible polynomial on $K[x]$
Let $F$ be a field and $a,b\in F$ with $a\ne0$. Then, $f(x)\in F[x]$ is irreducible if and only if $f(ax+b)\in F[x]$ is irreducible.
This is my proof
$(\Rightarrow)$ Suppose $f(x)=h(x)g(x)$ is ...
- 11
3
votes
2
answers
157
views
Using cyclotomic polynomials to show a polynomial is irreducible over $\mathbb{Q}$
I have been given the polynomial $$f(x)=x^8+x^7-x^5-x^4-x^3+x+1$$ and have been asked to show it is irreducible over $\mathbb{Q}$ by considering the product $(x^2-x+1)f$. (Looking it up, I realise $f$ ...
