# Questions tagged [irreducible-polynomials]

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

2,063 questions
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### If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Then $p(x) | f(x)$

If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Show: $$p(x) | f(x)$$ I'm not sure if it's a direct consequence of the proof that minimal polynomial of an ...
0answers
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### Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
0answers
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### $f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$ I think $p$ is supposed to be a prime for the only ...
1answer
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### Degree 2 Recurring monic polynomials

Consider a monic polynomial $x^2+ax+b=0$, with real coefficients. If it has real roots $p$ and $q$, such that $p\leq q$, then you construct a new monic polynomial as $x^2+px+q=0$. If this polynomial ...
1answer
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### Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
3answers
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### Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$ [duplicate]

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the ...
0answers
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1answer
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### when is the $n$-th cyclotomic polynomial irreducible over $\mathbb{R}$

It is well known that the cyclotomic polynomials $\Phi_n(x)$ are irreducible over the field of rationals $\mathbb{Q}$. I am curious about their reducibility over the real numbers $\mathbb R$. We have ...
1answer
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1answer
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### $y^2z - x^3$ is irreducible and with one singularity

Give an example for an irreducible cubic curve in $\mathbb{C}\mathbb{P}^2$ with exactly one singular point. It is easy to check that $y^2z - x^3$ has only [0,0,1] as a singularity. But how to show it ...
2answers
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4answers
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### Using Eisenstein criterion show that $x^3+x^2-2x-1$ is irreducible? [duplicate]

I tried for few primes but they do not satisfy Eisenstein criterion.Also is there any approach other than brute force with the help of which we can find that prime.
1answer
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### Show that a polynomial is irreducible in $\mathbb { Z } [ i \sqrt { 5 } ]$ but not in $\mathbb { Q } [ i \sqrt { 5 } ]$

To prove : Show that the polynomial is irreducible in $3 + 2 t + 2 t ^ { 2 }$ over $\mathbb { Z } [ i \sqrt { 5 } ]$ but not over $\mathbb { Q } [ i \sqrt { 5 } ]$. Solution : The roots of the ...
5answers
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### Showing $f(t) = t^4 + 2t^2 + 9$ is reducible over $\Bbb Q$

So, as a part of a homework problem, I'm tasked with showing that $f(t) = t^4 + 2t^2 + 9$, despite having no roots in the rationals, is reducible over them. The former task - showing a lack of ...
1answer
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### Existence of irreducible polynomials with certain criteria

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements, where $q$ is an odd prime power. The question is as follows: Does there exists $a\in \mathbb{F}_{q}^*\setminus (\mathbb{F}_{q}^*)^2$, such ...
2answers
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### Irreducibility of a simple polynomial

For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $\mathbb{Q}$. What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 \mod 4$, ...
1answer
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### Factorise $x^{111}+9x^{74}+27x^{37}+27$ in $\mathbb{Z}[x]$

I am having trouble with the following. Factorise $x^{111}+9x^{74}+27x^{37}+27$ in irreducible factors in $\mathbb{Z}[x]$. I did not find it to be an Eisentein polynomial and trying to find zeros ...
1answer
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### Direct sum factorization of polynomials

I have been recently reading the paper "Mixed finite elements for second order elliptic problems in three variables" by Brezzi et. al. I noticed the claim in the proof of Lemma $2.1$, which basically ...
1answer
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### Reducible and Irreducible polynomials over $\Bbb Q$

Consider polynomials of the form: $$x^r-(1-x)^k,$$ for $r,k\ge2.$ $x\in(0,1).$ When $r=k$ the polynomial seems to be reducible, except at $r=k=2.$ Do the irreducible and reducible polynomials ...