All Questions
186
questions
2
votes
2
answers
63
views
If $p(x) = a_3 + a_2x + a_1x^2 + a_0x^3$ is irreducible in $\mathbb{Q}[x]$, then $q(x) = a_0 + a_1x + a_2x^2 + a_3x^3$ is also irreducible?
I have this doubt because what I want to prove is that, with the given hypothesis,$\frac{\mathbb{Q}[x]}{\langle q(x) \rangle}$ is an integral domain. I have the same problem but for polynomials of ...
2
votes
1
answer
156
views
Are two principal ideals are equal if their generator has the same root in some extension?
I recently came across the following claim in a paper concerning polyonomials in $\mathcal{R}_p = \mathbb{Z}_p[X]/\langle X^d + 1\rangle$ (where $\mathbb{Z}_p = \mathbb{Z}/\mathbb{Z}p$) and their ...
- 147
-1
votes
1
answer
150
views
If a monic polynomial is irreducible in $\mathbb Z_{p^k}[X]$, then it is irreducible in $\mathbb Z_p[X]$
Let $p$ be a prime and $k$ is a positive integer. There is a natural projection of rings $Z_{p^k}[X] \longrightarrow Z_{p}[X]$. Suppose a monic polynomial $f(X) \in Z_{p^k}[X]$. $f(X)$ is called basic ...
- 1,201
3
votes
1
answer
190
views
Generalization of Eisenstein's Criterion [duplicate]
Let $f(X)=a_{2n+1}X^{2n+1}+\ldots+a_0\in \mathbb{Z}[X]$ with
$$\begin{align*}
a_{2n+1}&\not \equiv 0 \pmod p\\
a_{2n},\ldots,a_{n+1} &\equiv 0 \pmod p\\
a_n,\ldots,a_0&\equiv 0 \pmod{p^2} ...
- 2,234
0
votes
1
answer
80
views
Some properties about $\mathbb{F}_3[x]/(x^3+x+1)$
I am given $L:=\frac{\mathbb{F}_3[x]}{(x^3+x+1)}$ and I have to prove different properties about this object.
First of all, since the polynomial for which I make the quotient is reductible $$x^3+x+1=(...
- 657
0
votes
0
answers
96
views
Question on Gauss’s Lemma of Irreducibility
I am having trouble understanding what is wrong with my proof of the forward direction of this lemma.
My proof does not assume the natural embeddings involved. I specifically write them out because it ...
- 51
0
votes
2
answers
162
views
Irreducibility on homogeneous polynomials
I would like to ask you the following irreducibility.
Prove that $x^3 - xy^2 + y^3 \in \mathbb{Q}[x, y]$ is irreducible. I know that $t^3 - t + 1 \in \mathbb{Q}[t]$ is irreducible by Gauss's lemma. I ...
- 425
0
votes
1
answer
95
views
Factorize $(X+1)^{101}+100$ in irreducibles in $\mathbb{Q}[X]$.
As the title says I'm trying to factorize the polynomial $f(X)=(X+1)^{101}+100$ in irreducibles in $\mathbb{Q}[X]$. I got a hint stating that 101 is prime, but I don't see how that can be useful. ...
- 329
0
votes
1
answer
212
views
Show that $3x^5-4x^3-6x^2+6$ is irreducible over $\mathbb{Q}[x]$
Can’t seem to apply Eisenstein’s criterion. It doesn’t make sense to reduce modulo 2 since the constant term disappears… What are some other possible strategies to apply here?
- 91
0
votes
1
answer
60
views
Irreducibility of polynomial P(X) = X
I have the following exercise:
Let $R$ be the integral domain of all polynomials $P(X)$ with real coefficients whose constant term is a rational.
Is the poly $P(X) = X$ irreducible in $R$?
My question ...
- 57
0
votes
1
answer
80
views
Irreducible polynomials and factors
Let $f(x) \in F[x]$ where $F$ is a field. The remainder theorem states that the remainder when $f(x)$ is divided by $x-a$ is $f(a)$. This implies that $f(a) = 0$ if and only if $x-a$ is a factor. This ...
- 321
0
votes
0
answers
46
views
Irreducibility of a polynomial over the field of rational complex functions
Let $k=\mathbb{C}(t)$ be the field of rational functions over $\mathbb{C}$.
I want to show that $P(x)=x^2+t \in k[x]$ is irreducible over $k$, and further find the degree of the splitting field of $P(...
- 99
1
vote
1
answer
462
views
Irreducible polynomial in field of positive characteristic
Let $F$ be a field of characteristic $p > 0$ and $C$ be an element of $F$ that is not a $p$-power. For a positive integer $s$, show that $x^{p^s} − C$ is irreducible, and its splitting field is of ...
- 743
0
votes
1
answer
44
views
Irreducible Polynomial in $\mathbb{F}_p (T) [x]$ [duplicate]
Let p be prime, and $K = {\bf F}_p (T)$ an extension of ${\bf F}_p$. How can i prove that the polynomial $x^p - T$ is irreducible in $K[x]$
- 31
0
votes
2
answers
72
views
Difficulty understanding how $3x^2 + 12$ in $\mathbb{C}[x]$ is reducible.
I am having difficulty understanding how $3x^2 + 12$ in $\mathbb{C}[x]$ is reducible.
It can be factorized into $3(x^2 + 4) = 3[(x + 2i)(x - 2i)]$. However, isn't $3$ a unit in $\mathbb{C}[x]$? After ...
- 313