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Questions tagged [irreducible-polynomials]

Often called prime polynomials. Polynomials that have no polynomial divisors.

An irreducible or prime polynomial is a polynomial that has no polynomial factors. The parallel with prime integers is appropriate. In the same way that a prime number cannot be divided by any other integer other than $\pm1$, an irreducible polynomial cannot be divided by any polynomial other than a constant non-zero polynomial.

Though it is usually clear from context, care should be taken in observing the domain over which a polynomial is irreducible. For instance, $x^2+4x+13$ is irreducible over $\mathbb{R}$ but not $\mathbb{C}$; likewise, $x^3 + 2 x^2 + x + 3$ is irreducible over $\mathbb{F}_{5}$ but factors as $(x-1)(x-2)^2$ over $\mathbb{F}_7$.

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About a form of a real irreducible polynomial

Let $f(x_1,...,x_n)\in \mathbb R[x_1,...,x_n]$ be an irreducible (over $\mathbb R$) polynomial with real coefficients. Let $f$ has a factor in $ \mathbb C[x_1,...,x_n]$ of degree $\geq 1$. Is $f$ of the form $f=\alpha r\cdot c(r)$, where $\alpha \in…
Alex
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Existence of an irreducible trinomial over finite fields?

Let $F_q$ be the finite field of $q$ elements. I am looking for the existence of an irreducible trinomial of the following form: $$x^n-x^m-1$$ over $F_q$ for some $n,m.$ I think it should be true because we may choose $n$ big enough. In the case,…
Hieu
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Is the following polynomial irreductible over $\mathbb{Z}[X]$?

Is the following polynomial irreductible over $\mathbb{Z}[X]$? $f(x) = (x-11)(x-8)(x-2017)(x-17)(x+5)-1$ What I have tryied: Assume that there is a root a in $\mathbb{Z}$ then $f(a) = 0$ which implies that the polynomial…
Raducu Mihai
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Proving irreducibility of a polynomial

I'm trying to prove that the following polynomial $x^7+3x^6+12x^5+6x^4+2x^3-4x^2+6x+2$ is irreducible in $\mathbb{Q}[x]$. I began with using the reduction mod p (p a prime), using 5 as my prime, test to show that the polynomial is irreducible in…
Mo Gainz
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Is $1+X^2$ irreducible in $\mathbb{Z}_3$?

$N = 1 + X^2$ is irreducible in $\mathbb{Z}_3[X]$ , since $1+0^2 = 1 $ and $1+1^2 = 1 + 2^2 = 2$. Which means that $N$ can never be zero. Regarding the factor theorem, $N$ is irreducible. Is this sufficient to prove the irreducibility?
user1511417
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Irreducibility of a mulativariate polynomial after perturbation of constant

$\newcommand\C{\mathbb{C}}$ $\newcommand\bs{\backslash}$ I have several questions which are related to one question but for which I slightly change the original question. I believe the answer to all is yes, but I have not worked any of them…
quantum
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Checking absolute irreducibility

How can one check if a polynomial $f$ is absolutely irreducible ? (meaning that it is irreducible over the closure of the underlying field. Example: $y^{2} - x^{3}$ is absolutely irreducible ( the underlying field being rational numbers) while…
Abhiroop Sanyal
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Is determing whether a polynomial is irreducible easier if all coefficients are non-negative?

I wonder whether the question whether a polynomial in $\mathbb Q[x]$ is irreducible over $\mathbb Q$ is easier if all coefficients are non-negative. I found various sufficient conditions for a polynomial to be irreducible in the internet, but…
Peter
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Find monic irreducible polynomials in $Z_3[x]$ which divide both f(x) and g(x)

The Problem Find all monic irreducible polynomials in $Z_3[x]$ which divide both $f(x)$ and $g(x)$. Where $f(x) = x^5+x^3+x^2+x+2$, $g(x) = x^6+x^5+x^3+1$ This problem is part of an early exam so it comes with a solution. My attempt at a…
ejbs
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Alternative proof of irreducibility of a geometric series polynomial

Prove, in $ \displaystyle \mathbb{Z} \left[ x \right]$, the irreducibility of $ \displaystyle p + \sum_{k=1}^n x^k$ where $n>1$ and $p$ a prime number. It is not too difficult to exploit the magnitude of the roots to create a contradiction under the…
Jack Tiger Lam
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Which of the following is an irreducible factor of $x^{12}-1$ over $\mathbb Q$?

Which of the following is an irreducible factor of $x^{12}-1$ over $\mathbb Q$? $x^8+x^4+1$ $x^4+1$ $x^4-x^2+1$ $x^5-x^4+x^3-x^2+x-1$ Is the answer $(1)$ correct? I am not sure which one is.
SMM
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Let $f$ be a polynomial with $\deg f = 2n + 1$, and $a_1, .\dots, a_{2n+1}$ that $f(a_i) \in \{+1, -1\}$. Show $f$ is irreducible

I am facing the following problem: Let $n \in \mathbb{N}_{\geq0}$ and $f \in \mathbb{Z}[X]$ be a polynomial with $\deg f = 2n + 1$, as well as (in pairs different) $a_1, \dots, a_{2n+1}$ such that $f(a_i) \in \{+1, -1\}, \forall i \in…
johnnycrab
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Show that a polynomial of degree 2 is reducible if it has roots.

A polynomial $f\in k[x]$ is irreducible if : 1. $\deg(f)\geq 1$ 2. If $f=gh$ then $\deg(g)=0$ or $\deg(h)=0$ First of all, this definition confuses me. For example, Let $f=2x^2+2$ and the field be $\mathbb{R}$. By this definition f is irreducible…
NormalsNotFar
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If $P$ is monic with degree $d$, at least one of $P-1,P-2,\ldots ,P-(d+1)$ is irreducible over $\mathbb Q$

Let $P\in{\mathbb Z}[X]$ be a monic polynomial of degree $d>1$. When $d=2$ or $3$, it is easy to see that at least one of $P-1,P-2,\ldots,P-(d+1)$ is irreducible over $\mathbb Q$ (see below). Does this property still hold for $d \geq 4$…
Ewan Delanoy
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Irreducible polynomials with same root

If a monic polynomial $p(x)$ is irreducible over rationals and has $\alpha$ as root. Is it possible that $\alpha$ is root of some other irreducible monic polynomial $q(x)$ such that $\deg(p(x))\neq \deg(q(x))$? This question arose when I was…
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