All Questions
28
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 ...
0
votes
1
answer
160
views
$x^2-y^2-1$ irreducible in $K[x,y]$
Being $K$ any field, I am trying to figure out if $x^2-y^2-1$ is irreducible in $K[x,y]$.
My approach was to assume there is a decomposition $x^2-y^2-1=(ax+by+c)(ex+fy+g)$ and try to reach a ...
- 904
0
votes
1
answer
51
views
Examples of coprime polynomials $f(x)^3+g(x)^2=h(x)$ with $6\deg h<\deg g$
Let $k$ be a field s.t. $\mathop{\mathrm{char}}k\neq2,3$. Let $d\geq 1$, $f\in k[x]$ be a polynomial of degree $4d$, $g\in k[x]$ be a polynomial of degree $6d$ with $\gcd(f,g)=1$. So $f^3$ and $g^2$ ...
- 1,785
2
votes
1
answer
58
views
Irreducibility of a polynomial in $\mathbb{Q}[x,y]$ [closed]
Let $x^4+x^3y+x^2y^2+xy^3+y^4 \in \mathbb{Q}[x,y]$ be a primitive polynomial for which we have to investigate if it is irreducible. My idea is:
since $(y-1) \in \mathbb{Q}[x,y]$ is a prime ideal such ...
- 415
2
votes
0
answers
54
views
Hilbert's Nullstellensatz and irreducible polynomials [duplicate]
In p.35 of Miranda's book Algebraic Curves and Riemann Surfaces, the following theorem is stated and is called as Hilbert's Nullstellensatz.
Theorem. Let $f\in \Bbb C[x_1,\dots,x_n]$ is an irreducible ...
- 1,802
2
votes
2
answers
323
views
If I erase one of the variables in an irreducible homogeneous polynomial, it's still irreducible
If $f$ is an irreducible homogeneous polynomial in $k[x_0,...,x_n]$ then $f$ is supposed to cut out an irreducible hypersurface in $\mathbb{P}_k^n$. So if I look in an affine chart, I should see an ...
2
votes
1
answer
371
views
Showing that $y$ is irreducible in $\mathbb R[x,y]/(x^2+y^2-1)$.
Let $B = \mathbb R[x,y]$ where $x^2 + y^2 = 1$ which is called the coordinate ring of the unit circle.
I am trying to prove that $y$ is irreducible in $B.$
I have the following information about $B$:
...
user889696
0
votes
1
answer
183
views
Polynomial $f(X) g(Y)- g(X) f(Y)$ irreducible in $F[X,Y]$?
Let $F$ be a field and $f(X), g(X) \in F[X]$ two coprime
non constant polynomials.
Question: Is the polynomial $P(X,Y):= f(X) g(Y)- g(X) f(Y)$
irreducible in the ring $F[X,Y]$ of polynomials in two ...
- 7,499
0
votes
1
answer
108
views
Showing that a polynomial in $k[x_1, \dots, x_n]$ is irreducible
Let $p\in k[x_1, \dots, x_n]$ such that p does not involve the variable $x_i$ and is not square of any polynomial. Then we want to show that the polynomial $f = x_i^2-p$ is irreducible.
I tried ...
- 518
1
vote
1
answer
57
views
Checking irreducibility of polynomials in two variables
There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy.
...
- 808
3
votes
0
answers
237
views
Special irreducible polynomials in $k[x,y]$
Let $k$ be a field of characteristic zero, $n \in \mathbb{N}$.
Definitions:
(1) $0 \neq f \in k[x_1,\ldots,x_n]$ is always irreducible, if for every $\lambda \in k$, $f+\lambda$ is irreducible in $...
- 6,705
1
vote
0
answers
77
views
Different formulations of Gauss's lemma
This may be a somewhat dumb question, but I have recently learned of Gauss's lemma, from 2 different books, that give 2 different formulations and I must be having some kind of mental block, because I ...
- 1,434
2
votes
2
answers
139
views
Monic polynomials $f\in R[x]$ with $f-a$ irreducible for almost all values of $a\in R$
I assume all the rings to be commutative and unital. Recall that an element $b$ of a ring $S$ is irreducible if $b\notin\{0\}\cup S^*$ and $b$ is not the product of two noninvertible elements in $S$.
...
- 6,131
0
votes
0
answers
98
views
Weak Nullstellensatz in practice
Let $k$ be an algebraically closed field (I do not mind to assume $k=\mathbb{C}$).
Let $f(x),g(x) \in k[x]$.
Hilbert's weak nullstellensatz says that exactly one of the following statements holds:
(1)...
- 6,705
0
votes
0
answers
482
views
Hessian of a smooth homogeneous polynomial of degree $d \ge 3$ is homogeneous of degree $3(d-2)$?
Let $k$ be an algebraically closed field of characteristic zero, let $f(X,Y,Z) \in k[X,Y,Z]$ be a homogeneous polynomial of degree $d \ge 3$ such that $\Big(\dfrac {\partial f}{\partial X} (p) ,\dfrac ...
- 4,454