Intereting Posts

Prove that if $f$ is integrable on $$ then so is $|f|$?
convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$
Prove that a covering map is a homeomorphism
Beta function derivation
Any more cyclic quintics?
Isomorphism between quotient modules
Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?
Is length adimensional when space is not flat?
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
Identifying the two-hole torus with an octagon
Optimal control
If $x$ is a positive rational but not an integer, is $x^x$ irrational?
Every finite abelian group is the Galois group of of some finite extension of the rationals
The smallest integer whose digit sum is larger than that of its cube?

Facts before question:

$\textbf{Fact 1:}$ Let $F(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$.

If $|a_{n-1}|>1+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ irrreducible in $\mathbb{Z}[X]$.

- Irreducibility of $X^{p-1} + \cdots + X+1$
- How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
- How many irreducible factors does $x^n-1$ have over finite field?
- Number of monic irreducible polynomials of prime degree $p$ over finite fields
- How can I show that the polynomial $p = x^5 - x^3 - 2x^2 - 2x - 1$ is irreducible over $\Bbb Q$?
- Irreducible polynomial which is reducible modulo every prime

$\textbf{Fact 2:}$ Let $K$ be a field, $F(X,Y)=a_n(X)Y^n + \cdots + a_1(X)Y + a_0\in K[X,Y]$, with $a_o,\ldots,a_{n-1}\in K[X]$, $a_n \in K$ and $a_0a_n\neq 0$.

If $\deg(a_{n-1})>\max\bigl(\{\deg(a_0), \deg(a_1), \ldots, \deg(a_{n-2})\}\bigr)$, then $F$ is irreducible over $K[X]$.

Facts end here.

$\textbf{Conjecture:}$ Let $F(X) = a_nX^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$.

If $|a_{n-1}|>|a_n|+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ irrreducible in $\mathbb{Z}[X]$.

Is the above conjecture true?

I know nothing about multivariable polynomials, so I’m just looking for a yes or no answer, and, in case it is false, can you please provide a counterexample?

- Revisited: Binomial Theorem: An Inductive Proof
- Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots
- $\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$
- What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?
- Finding the all roots of a polynomial by using Newton-Raphson method.
- Factoring $x^6 + x^4 + x^3 + x + 1$ over $\mathbb{F}_{16}$
- Finding a common factor of two coprime polynomials
- If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$
- Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q$.
- How to check that a cubic polynomial is irreducible?

Try this for the polynomial $2X^2 + 5X + 2$.

There shouldn’t be an absolute value of $a_n$ in there. The polynomial must also be monic.

- What infinity is greater than the continuum? Show with an example
- Why isn't $\mathbb{C}/(xz-y)$ a flat $\mathbb{C}$-module
- Show group of order $4n + 2$ has a subgroup of index 2.
- A zero sum subset of a sum-full set
- A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
- Donsker's Theorem for triangular arrays
- Minimum Modulus Principle for a constant fuction in a simple closed curve
- Trying to get a bound on the tail of the series for $\zeta(2)$
- Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$
- Finite group is generated by a set of representatives of conjugacy classes.
- Prove that $e^{-A} = (e^{A})^{-1}$
- Poisson random variables and Binomial Theorem
- Formula for the harmonic series $H_n = \sum_{k=1}^n 1/k$ due to Gregorio Fontana
- Concept behind the limit to infinity?
- In what sense is the derivative the “best” linear approximation?