Questions tagged [irreducible-polynomials]

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

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if $ax^2+bx+c$ and $2ax+b$ expressions have a common divisor, then $ax^2+bx+c$ is a full square.

if $ax^2+bx+c$ and $2ax+b$ expressions have a common divisor, then $ax^2+bx+c$ is a full square. I would like to prove this statement. Thank you for any idea.
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Irreducible factors of minimal and characteristic polynomial of a endomorphism over a finite dimensional $\mathbb{F}$-vector space [duplicate]

Let $V$ be a finite dimensional $\mathbb{F}$-vector space. Suppose $L:V\to V$ is an endomorphism, whose associated matrix is $A$. Now, denote its characteristic and minimal polynomial by \begin{align*}...
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Homogenous polynomials and irreducibility

Hi i'm trying to do an exercise of algebra and i'm stuck. Here it goes Let $A$ be a factorial ring, $F_n$ and $F_{n+1}$ two non-zero polynomials of $A[X_1,\cdots ,X_k](k\in \mathbb{N}^{*})$, ...
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Question about irreducible polynomials in algebraically closed fields

A tipical question in the past exams of a commutative algebra course is to determine if a ring of the form $k[x_1,\dots,x_n]/f(x_1,\dots,x_n)$ is an integral domain, usually with $1\le n \le 3$. ...
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Eisenstein criterion and Newton polygon

Eisenstein criterion of irreducibility over $\Bbb Q_p$(so ofcourse over $\Bbb Q$) is also proved by using Newton polygon. The proof goes like this, Let $f(x)＝a_nx^n＋･･･＋a_1x＋a_0$ be $p$-Eisenstein ...
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How to tell if a multivariable polynomial is irreducible?

If I am given a multivariable polynomial $F(x_1,...,x_n)$ with real coefficients, how do I tell if it is irreducible? Similarly, I would like to be able to do this for polynomials with complex ...
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Is $(p^{\frac{p^2+1}{p^5-1}}-x^{p^3})^{p^2}+p^{p^2+1}x-p^{p^2+\frac{p^2+1}{p^5-1}}$ eisenstein?

I could not decide whether the following polynomial is $$f(x)=(p^{\frac{p^2+1}{p^5-1}}-x^{p^3})^{p^2}+p^{p^2+1}x-p^{p^2+\frac{p^2+1}{p^5-1}}$$ an eisenstein polynomial? If I write simply \begin{...
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Prove the irreducibility of $P(x)$ which satisfies: $xP(x-1)=(x-2022)P(x)+2022$

Prove the irreducibility in $\mathbb{Z}[x]$ of $P(x)$ which satisfies: $xP(x-1)=(x-2022)P(x)+2022, \forall x\in \mathbb{R}$ My attempts I thought of substituting $x$ for $0$ and $x$ for $2022$, but ...
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Factor $X^7 − 1$ into irreducibles in $\mathbb{Z}_{127}[X]$

Problem: Factor $X^7 − 1$ into irreducibles in $\mathbb{Z}_{127} [X]$ I'm having a lot of trouble understanding how to approach this, and this course topic in general. Any help (or even better, a ...
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Romania TST 1995 -irreducible Polynomail

Question is: Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and ...
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Checking if given polynomials are units in $\mathbb{Z}_7[x]$ [duplicate]

So, I was doing an algebra exercise related to $\gcd$'s and $PID$'s and I need to check if some polynomials in $\mathbb{Z}_7[x]$ are units. The polynomials are the following: \begin{equation*} g = x^2+...
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Separable and irreducible polynomial over $F$

Let $f(x) \in F[X]$ a polynomial with degree 5 and Galois Group $S_5$. Show that $f(x)$ is separable and irreducible over $F$. For irreducibility, I tried by contradiction $$f(x) = g(x) * h(x)$$ ...
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Showing $f(x) = x^4 - 6x^2 - 8x + 3$ is irreducible in $\mathbb{Q}[x]$ [duplicate]

I wish to determine that $f(x) = x^4 - 6x^2 - 8x + 3$ is irreducible in $\mathbb{Q}[x]$ by substituting $x + m$ for $x$ and then choosing $m$ appropriately. $\textit{Work so far:}$ By rational root ...
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Irreducibility of $x_1^2+x_2^2+...+x_n^2$ in $\mathbb{C}[x_1,...,x_n]$ [duplicate]

I wanted to know whether the polynomial function $x_1^2+...+x_n^2 \in \mathbb{C}[x_1,...,x_n]$ is irreducible in $\mathbb{C}[x_1,...,x_n]$ for $3 \leq n$ ?
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How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreductible in $\mathbb{F}_{2}$

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreducible over $\mathbb{F}_{2}$. The main 2 "weapons" I have at my disposal is the Eisenstein criteria and reduction criteria but neither seem to ...
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Irreducible polynomials in $\mathbb Z[X]$ with no primes in the image.

It seems intuitive that if an irreducible polynomial $q\in\mathbb Z[X]$ has no primes in it's image, i.e. $n\in\mathbb Z\implies q(n)\notin \mathbb P$, then there is a non unit integer $m\in\mathbb Z$ ...
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Minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$

I was asked to calculate the minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$ over the fields \begin{align*} K_1 = \mathbb{Q}, \quad K_2 = \mathbb{Q}(\sqrt{2}), \quad K_3 = \mathbb{Q}(i\sqrt{3}), \...
Let $\ a,b\$ integers with $\ 0<a<b\$. Consider the following family of polynomials : $$f(x)=x^b+x^a+1$$ $$g(x)=x^b+x^a-1$$ $$h(x)=x^b-x^a+1$$ $$i(x)=x^b-x^a-1$$ The problem is about which of ...
Divisibility in $\mathbb{Q}[x]$
Let $F(x),G(x) \in \mathbb{Z}[x]$ with $G(x)$ being irreducible in $\mathbb{Z}[x]$. (By irreducible, we mean that if $G(x)=A(x)B(x)$ for $A(x),B(x) \in \mathbb{Z}[x]$, then either $A(x)$ or $B(x)$ is ...