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Questions tagged [irreducible-polynomials]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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 $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
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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 ...
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Polynomial that has no rational roots yet has roots modulo every integer

I want to characterize when the polynomial $p(x) = (x^{2} - a) (x^{2} - b) (x^{2} -ab)$ has a root modulo every integer, yet doesn't have an integer root. I worked out the condition when '$a$' and '$...
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Proving $f(x)=x^4+4x^3+3x^2+7x-4$ is irreducible

Here's my attempt, I'm almost there but I'm stuck: Using a hint, I wrote the modular reduction: Reducing the coefficients modulo $2$ gives: $\left [ f \right ]_2=x^4+x^2+x=x(x^3+x+1)$. Reducing ...
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How to show a polynomial is reducible/irreducible in a ring

I have to show that these polynomials is reducible or irreducible in the given ring. $a)$ $2x^3 − 5x^2 + 6x − 2$ in $\mathbb{Z}[x]$ $b)$ $x^4 + 4x^3 + 6x^2 + 2x + 1$ in $\mathbb{Z}[x]$ I think I ...
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Proof that a polynomial is irreducible over Q [duplicate]

How could I prove that $f=[(X-1)(X-2)...(X-n)]^{2}+1$ is irreducible over $\mathbb{Q}[X]$ ? I've tried to use Pólya's query of divisibility: If $f$ in $\mathbb{Z} [X]$ is a polynomial of degree $...
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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 ...
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Family of irreducible polynomials

Consider the following family of polynomials $$P_n(X)=\sum_{i=0}^n(n+1-i)X^i,\,n\ge 1$$ Let’s write down the first few $$ \begin{align} P_1(X)=&X+2\\ P_2(X)=&X^2+2X+3\\ P_3(X)=&X^3+2X^2+...
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Prove that $x^2 − 2$ is irreducible over $\mathbb Q (\sqrt 3)$

Prove that $x^2 − 2$ is irreducible over $\Bbb Q(\sqrt 3)$. I was originally trying to use the fact that if $K=\Bbb Q(\sqrt 3)[x]/(x^2-2)$ and $[K:\Bbb Q(\sqrt 3)]=2$ then $x^2-2$ is irreducible over ...
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How to prove irreducibility of this polynomial?

Let $p,q$ be primes. Prove that $y^{n }-p$ is irreducible over $\mathbb{Q}(\sqrt[n ]q) $ . I have tried for some time, but still feel confused about how to prove it. Can anyone help me?
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How do I show that $y^2 - x(x-1)(x+1)$ is irreducible in $\mathbb{R}[x, y]$?

I tried writing this polynomial as a product of two polynomials $g, h$ of degree 2 and 1, respectively, and tried to arrive at a contradiction by multiplication of their coefficients. However, I ...
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4answers
86 views

Irreducible Polynomial Field Extensions with Root $\cos \frac{2\pi}{7}$

Show that $\theta = \frac{2k\pi}{7}$ satisfies the equation $\cos 4\theta − \cos 3\theta =0$ for each integer $k$. Hence find an irreducible polynomial over $\Bbb Q$ with $\cos \frac{2\pi}{7}$ as a ...
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Show that the polynomial $7X^5 + 71X^3 - 9$ is irreducible in $\mathbb{Z}[X]$

Show that the polynomial $f(X) = 7X^5 + 71X^3 - 9$ is irreducible in $\mathbb{Z}[X]$ My solution: Using the irreducibility test: "Reduction Mod p Test" $f(X)$ is clearly primitive and that the ...
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Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible

Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$...
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The real positive root of $9x^5+7x^2-9=0$

$$9x^5+7x^2-9=0$$ How do we evaluate the roots of the given polynomial? We're asked to find its real positive zero. What I tried doing: Let $$f(x)=9x^5+7x^2-9$$ Using Descartes' rule of signs, I ...
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Irreducible polynomial with root implies least degree

Let $\alpha$ be an element of an extension field $K$ of the field $F$ that is algebraic over $F$. Then, if $f$ is an irreducible element of $F[x]$, and $\alpha$ is a root of $f$, then $f$ is the monic ...
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Irreducibility of $2x^3-3x^2+6$ in $\mathbb Q[x]$

The following statement is given, i need to check whether its true\false. There exists a subfield $F$ of $\mathbb{C}$ such that $F\not\subseteq\mathbb R$ and $$F \cong \mathbb Q[X]/(2X^3 − 3X^2 + 6)....
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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 ...
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51 views

Show some polynomial satisfies Eisenstein's Criterion

Consider a polynomial $$f(X) = X^{(p-1)p^{n-1}} + X^{(p-2)p^{n-1}} + \cdots + X^{p^{n-1}} + 1$$ Now I need to show $f(X+1)$ safeties Eisenstein's criterion. My argument is that $$f(X+1) = (X+1)^{...
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Irreducibility of polynomials over some quadratic fields

Prove the following statements: a) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{2})$ b) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{-3})$ c) The polynomial $x^3-x-...
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Is $x^3-1$ reducible over $\mathbb{Q}$ [duplicate]

In other discussions, e.g. here it is claimed that a polynomial in a field $K$ with degree greater than $1$, having a root in $K$, must be reducible. So by this criterion $X^3-1$ would be reducible ...
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Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
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$f(x) = x$ in $(\mathbb{Z}/6\mathbb{Z})[x]$ factors as $(3x+4)(4x+3)$

This question has a lot of parts so I'll post each part separately. First, show $f(x) = x$ in $(\mathbb{Z}/6\mathbb{Z})[x]$ factors as $(3x+4)(4x+3)$ I am trying long division. I cannot divide $x$...
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Units, Primes and Irreducibles

How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$? Thank you.
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Degree of splitting field of $X^4+2X^2+2$ over $\mathbf{Q}$

Find the degree of splitting field of $f=X^4+2X^2+2$ over $\mathbf{Q}$. By Eisenstein, $f$ is irreducible. By setting $Y=X^2$, we can solve for the roots: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
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Degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$

What is the degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$ ? I note that $\cos(2\pi/8) + i \sin(2\pi/8)$ is a root of $x^8-1$. $x^8-1$ can be factored into $x^8-1 =(x^4+1)(x^2+1)(x+1)(...
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Check the properties of given polynomial of order $7$

Consider $$f(x)=x^{7}-105x+12$$ Then which of the following is\are correct? $f(x)$ is reducible over $\Bbb Q$ There exists an integer $m$ such that $f(m)= 105$ There exists an integer $m$...
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Is $x^4+4x-1$ irreducible in the field $Q[\sqrt{-7}]$?

I already know this is irreducible over $\mathbb{Q}$. Right now, my general plan is to show there is no root in $\mathbb{Q\sqrt{-7}$ (thus no linear factor). And then show that two quadratic factors ...
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1answer
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Show that $x^p - m$ is irreducible for prime $p$ and $m \in \mathbb{Q^{\times}}\setminus \left(\mathbb{Q^{\times}}\right)^p$ [duplicate]

I'm stuck if $m$ is not a prime or has a single prime divider (Then using Eisenstein's criterion), e.g, $m=4$ and $p=5$. Any suggestions?
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1answer
28 views

Units, Prime Elements and Irreducible elements for polynomials

Are there any general steps or things to look out for when identifying the unit elements, irreducible elements and prime elements in ring of polynomials e.g. $\mathbb{Z}_4[x]$, $\mathbb{C}[x]$, $\...
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2answers
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If a monic rational polynomial of degree $p-1$ has $p$-th root of unity as a root, is it the cyclotomic polynomial?

If a monic rational polynomial of degree $p-1$ has a $p$-th root of unity as a root, where $p$ is prime, does that make it the cyclotomic polynomial $x^{p-1}+...+1$? I think this is the same as ...
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Factoring of polynomial over the rationals (quick question)

Okay this might be a stupid question, but couldn't find any clear explanation on the internet, so here goes. If I factor a polynomial into products of irreducible polynomials, are the products of ...
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1answer
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Irreducible Polynomials In $F_3$

Let us say I have some irreducible polynomials in $F_3$ $$p(x) = x^3 + 2x + 2$$ and $$p(x) - 1 = x^3 + 2x + 1.$$ Now, using the power of Maple and Wolfram Alpha, we can check that $$p(x^{13}) = x^{...
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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.
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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 ...
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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 ...
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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 ...
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2answers
44 views

polynomial units in field

I am new on this topic and would like to have some clarifications. We were given a definition and theorem as follows: Definition : A polynomial $p$ in $F[x]$ is irreducible if $p$ is not a unit of $...
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Why is $(X^5-1)/(X-1)$ the minimal polynomial for $e^{2\pi i/5}$, the fifth root of unity, over $\Bbb Q$?

I have some trouble understanding an example from my reader$^1$. We have the fifth root of unity $\zeta_5=e^{2\pi i/5}$, so by definition $\zeta_5^5=1$. Then I want to find its minimal polynomial ...
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1answer
117 views

Prove that $f(x)$ and $g(x)$ do not have any roots in common.

Suppose that $a(x)f(x) +b(x)g(x) = 135$ where $a(x), b(x), f(x)$ and $g(x)$ are polynomials over $F$. Prove that $f(x)$ and $g(x)$ do not have any roots in common. Any help is appreciated; thanks!
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1answer
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Maximal ideals of $\Bbb F_2[x]$

Prove or Disprove: $\Bbb F_2[x]$ has uncountably many maximal ideals For every integer $n$, every ideal of $\Bbb F_2[x]$ has only finitely many elements of degree $\leq n$. The first ...
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1answer
79 views

Proving that there are infinitely many irreducible polynomial in $\mathbb{Z}_{5} [x]$ [duplicate]

Pinter's textbook "A book of abstract algebra" asks to prove the following: There are infinitely many irreducible polynomials in $\mathbb{Z}_{5} [x]$. Here's my attempt to prove it any field $\...
3
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2answers
218 views

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 $1 \mod 4$, ...
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1answer
51 views

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 ...
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1answer
36 views

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 ...
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1answer
75 views

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 ...