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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 $\mathbb{F}_7$.

3233 questions
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Why is $y^2-x^3\in \mathbb{C}[x,y]$ irreducible?

How can I prove that $y^2-x^3\in \mathbb{C}[x,y]$ is irreducible?
Nat
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When is the polynomial $X^a+Y^b \in \mathbb{Q}[X,Y]$ irreducible?

I am studying the irreducibility of the polynomial $X^a+Y^b$. I proved that if $X^a+Y^b \in \mathbb{C}[X,Y]$, the irreducibility is equivalent to that $a$ and $b$ is relatively prime. I also proved that $X^a+Y^b \in \mathbb{R}[X,Y]$, it is…
certain_integral
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Why $y^2-x^3-x^2$ is irreducible?

Let $p(x,y)=y^2-x^3-x^2$ a polynomial in $\mathbb{C}[x,y]$. How can I prove that $p(x,y)=y^2-x^3-x^2$ is irreducible as element of $\mathbb{C}[x,y]$?
ArthurStuart
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Irreducibility of an infinite sequence of polynomials

Prove that $P_n(X) = X^n - X^{n-1} - X^{n-2} - ... - X - 1$ is irreducible over $\mathbb{Z}$ for all $n$. I was able to prove the result for $n=2^k-1$ by applying Eisenstein's criterion to $P_n(X+1)$. But for other values of $n$, I'm stuck. Has…
Jean-Claude Arbaut
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The irreducibility of $x^{16}-2x^8+8x+1$ over the rationals

Question: Is $f(x)=x^{16}-2x^8+8x+1$ irreducible over the rationals? My attempt: Consider $f(x-1)$ which has every term (except for the highest and constant) divisible by $2$. To apply Eisenstein's criterion, I need to show that the constant term…
Paul
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Minimal degree of polynomial with $a^{1/b}$ as root

What is the minimal degree of a polynomial with integer coefficients with $a^{1/b}$ as a root where a and b are integers? Assume it can't be simplified, as in, there are no integer $x$ and $y
Alice Ryhl
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Irreducible polynomials of degree 3 and 5

Two part question: 1) There are three irreducible cubic polynomials $\mathbb F_2$ Aproach: Bruteforce, there are not as much cubic polynomials, just 8 of them: $x^3$ $x^3+1$ $x^3+x$ $x^3+x+1$ $x^3+x^2$ $x^3+x^2+1$ $x^3+x^2+x$ $x^3+x^2+x+1$ Only…
Timo Junolainen
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Proof that a Polynomial is Irreducible

In my abstract algebra class, we were given some polynomials to prove irreducible or not over the rationals. I have come up with a proof for this one, but I'm not sure if it's valid. The polynomial: $$x^4 + 2x^3 + x^2 + x + 1$$ I've started by…
NabiNaga
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Study irreducibility of $f(x) = x^{20} + 5x^{15}+25x^{10}+125x^5+625$ in $\mathbb Q[x]$

I started with the idea of looking for a $b \in \mathbb Z$ where $f(x+b)$ is a polinomyal that I can use Einsestein for $p = 5$. According with $\dbinom{5}k$ are multiples of $5$ I just need the independent term dividing $5$ and not $5^2$. Lets…
Spvf
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Proving a polynomial divides other polynomials

Given $\mathbb{F}_p[u]$ over $\mathbb{F}_p$ with $p$ prime, how can I prove that $u^2-cu+d$ irreducible is a factor of both $u^p+u-c$ and $u^{p+1}-d$?
frobeniusguy
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Irreducibility of $f(x)=x^n-p^m$

I have a question. Let $f(x)=x^n-p^m$ where $m$ and $n$ are coprime and $p$ is prime integer. How can I show that $f(x)$ is irreducible over $\Bbb Q$?
user152395
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Can every polynomial be moved to a irreducible one?

Given a polynomial $P(x)$ in $Q[x]$, can I always find a rational number, $c$, so that $P(x)+c$ is irreducible in $Q[x]$?
user157036
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$f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible?
Hesam
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Prove that : $x^n+4$ is irreducible in $\mathbb{Z}[x]$.

Prove that : $x^n+4$ is irreducible on $\mathbb{Z}[x]$ if only if $n\neq 4k$ with $k\in\mathbb{N}$.
Hung Nguyen
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Polynomials $p(x^k)$ with an irreducible factor of bounded degree.

Characterize the polynomials $f\in\mathbb{Z}[x]$ with the following property: There is $M\in\mathbb{N}$ such that for all $k\in\mathbb{N}^*$ there is an irreducible factor (over $\mathbb{Q}$) of $f(x^k)$ that has degree at most $M$. If $x\mid…
plop
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