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Questions tagged [irreducible-polynomials]

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

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2answers
45 views

Prove that $R$ is not a UFD [on hold]

Let, $$R=\{a_0+a_1x+\dots+a_nx^n \in \mathbb{Q}[x] : a_0 \in \mathbb{Z} , n \in \mathbb{N_0} \}$$ Prove that $R$ is not a UFD. I tried to find some irreducible element which is not prime but ...
3
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1answer
33 views

A computational criterion of irreducibility in $\mathbb Z[X]$?

If $f\in\mathbb Z[X]$ and there are $x_1,\dots, x_n\in\mathbb Z_+$, where $n>\deg f$, such that $f(x_i)\in\mathbb P$, $i=1,\dots n$, then $f$ is irreducible over $\mathbb Z$. Because, if $\,f=g\...
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1answer
19 views

If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$.

If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$. Is the statement ...
2
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1answer
26 views

Are the prime cyclotomic polynomials irreducible over any field where they're not obviously reducible ?

My question is the following : if $p$ is a prime number, $\Phi_p = \frac{X^p-1}{X-1}$, is $\Phi_p$ irreducible over any field $K$ where it has no root ? Phrased differently, if $K$ is of ...
1
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1answer
49 views

Quotient $\mathbf{F}_3[X]/(X^5+1)$

Factor $X^5+1\in\mathbf{F}_3[X]$ into irreducibles. What does the quotient $\mathbf{F}_3[X]/(X^5+1)$ look like? Since $-1$ is a zero, we divide $X^5+1$ by $X+1$ using long division, to obtain $X^5+1=(...
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0answers
13 views

Irreducibility of $x^p-x-a$ [duplicate]

Suppose I consider the polynomial $x^p-x-a$ over a field $k$ of characteristic $p$, how does one show that either the polynomial is irreducible or it splits into linear factors? It is clear to me that ...
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0answers
39 views

Is it true that he polynomial $\frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $\mathbb{F}_2[x]$ iff $p$ is prime?

Is it true that the polynomial $\frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $\mathbb{F}_2[x]$ iff $p$ is prime? I know it will be true in $\mathbb Q[x]$. Can anyone please help me to ...
1
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1answer
46 views

$x^{2n} + x^{2n-1} + x^ {2n-2} +\ldots+ x + 1$ is irreducible for any $n\in \mathbb N$ in $F_2[x]$. True or false?

Will the polynomials of the following set $A$ be irreducible in $F_2[x]$? $A = [x^{2n} + x^{2n-1} + x^ {2n-2} + \ldots+ x + 1 : n\in \mathbb N]$ Can anyone please give me hints how to proceed? ...
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1answer
46 views

Can we find all the irreducible polynomials of $F_2[x]$?

Can we find all the irreducible polynomials of $F_2[x]$ of a degree $n$? Is the number of irreducible polynomial of $F_2[x]$ Infinite? I was to find if there is any degree $n$ such that there is no ...
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0answers
39 views

Show $ \sum_f \lambda(f)t^{\deg f} = \prod_f \big(1 - \lambda(f)t^{\deg f}\big)^{-1} $

So I want to show that $$ \sum_f \lambda(f)t^{\deg f} = \prod_f \big(1 - \lambda(f)t^{\deg f}\big)^{-1} $$ where the sum is over monic polynomials and the product is over all monic irreducible ...
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0answers
18 views

Is the notion of irreducubility of polynomials defined only over fields, or can it be any set?

On Wikipedia, it only talks about a polynomial being irreducible over a field. But what if I want to talk about, say, a quadratic with integer coefficients whose factors consist of non-integers. Can I ...
5
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1answer
80 views

Why is $x^4 - 4x + 2$ irreducible over $\mathbb Q(i)$?

This is an exercise from Garling's A Course in Galois Theory Ch. 5 on irreducible polynomials. The other part of the question was to show that $x^5 - 4x + 2$ was irreducible over $\mathbb Q(i)$. I ...
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5answers
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Let k be a finite field. Is it true that the number of irreducible polynomials in k[x] is also finite?

I know this question has been asked before and I understand that it can be proved using the same sort of proof as the one used to show that there's infinite primes, but are there other ways of showing ...
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1answer
33 views

Question about constructing fields

Find all the monic irreducible polynomials in $F_5[x]$ of degree two (aside from $x^2-2$ and $x^2-3$, there are eight of them) Adjoining a root u of these polynomials to $F_5$, construct eight ...
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1answer
60 views

What does $φ(a) = a$ mean in this statement?

Let $F$ be a field and let $φ:F[x] \to F[x]$ be an isomorphism such that $φ(a)=a $ for every $a$ in $F$. Prove that $f(x)$ is irreducible in $F[x]$ if and only if $φ(f(x))$ is. [Hint: First prove that ...
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3answers
43 views

Let $f(x)$ be irreducible in $F[x]$, where $F$ is a field. If $f(x) | p(x)q(x)$, prove that either $f(x) | p(x)$ or $f(x) | q(x)$.

Let $f(x)$ be irreducible in $F[x]$, where $F$ is a field. If $f(x) | p(x)q(x)$, prove that either $f(x) | p(x)$ or $f(x) | q(x)$. Is this true because every irreducible element in a field is prime?
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0answers
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factors of $X^{p^n}-X$

I'm my study of Galois theory I have been struggling with the following proposition without much success: The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible ...
2
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1answer
64 views

Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.

Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$. Goal: Prove that $p$ is irreducible. Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by $$I:=(p).$$ My approach is to show that ...
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2answers
33 views

Irreducible polynomial over $\mathbb{Q}[x]$

Question: Check if: $$ f(x) = x^4 + 4x^3 + 6x^2 + 2x + 1 $$ is reducible or irreducible over $\mathbb{Q}[x]$ My Answer [Edited]: Suppose that $f(x)$ is irreducible over $Z_p[x]$ for a prime $p$, ...
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0answers
26 views

Proving an isomorphism between field extensions

I am trying to prove the following: Two field extensions $\mathbb{Q}[\sqrt{a}]$ and $\mathbb{Q}[\sqrt{b}],$ where $a,b \in \mathbb{Q}^\times,$ are isomorphic if and only if $\sqrt{a/b} \in \mathbb{...
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1answer
27 views

elliptic curve over GF(2^3)

This is $f(x) = x^3+x+3$ over $GF\left(2^3\right)$ How to know numbers of points on this equation? How to find those points? Is it an irreducible polynomial?
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1answer
45 views

What is the inverse of X modulo $1 + X + X^2 + X^3 + X^4$? [duplicate]

What is the inverse of X modulo $1 + X + X^2 + X^3 + X^4$? Is there any open softwares to calculate such things easily?
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1answer
19 views

Well definedness of multiplicity of a polynomial.

Let $F$ be a field. Let $a \in F$ be a root of $f(x)$. The multiplicity of $a$ is the maximum positive integer $m$ such that $(x-a)^m |f(x)$. I know that $x-a$ is an irreducible. How does this imply ...
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0answers
24 views

Fast modular composition and irreducibility testing

Given a polynomials $f\in F_q[t]$ we want an algorithm that says if $f$ is irreducible or not. In the book "Modern Computer Algebra" by von zur Gathen and Jürgen Gerhard, an algorithm for that is ...
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2answers
60 views

Compute the irreducible polynomials over Q for $a=\sqrt{2}+\sqrt{5}$ and $b=\sqrt[3]{2}+\sqrt{5}$

Compute the irreducible polynomials over Q for $a=\sqrt{2}+\sqrt{5}$ and $b=\sqrt[3]{2}+\sqrt{5}$ For a, I do: $$a=\sqrt{2}+\sqrt{5} \Rightarrow a^2-7=2\sqrt{10} \Rightarrow a^4-14a^2+9=0$$ So $p(X)...
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1answer
50 views

Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $\mathbb{Q}$? [duplicate]

Can I use Eisenstein's criterion to show $x^{4}-2x^{3}+2x^{2}+x+4$ is reducible over $\mathbb{Q}$? Can I say that there does not exist a prime that divides both $2$ and $1$? Or is there another way ...
2
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3answers
51 views

$x^3-2x-2$ is irreducible over $\mathbb{Q}$

I tried doing this by Eisenstein's criterion: $2$ is prime in $\mathbb{Q}$ and I then proceeded to write that it divides $-2$, $-2$ and $0$ but doesn't divide $1$ and also $2^2=4$ doesn't divide $2$. ...
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3answers
47 views

Unit in an Integral Domain $R$ implying a polynomial is irreducible in $R[X]$

I saw this theorem in another post with the comment that it is "easy to prove", and yet I'm struggling to see how it's simple. Theorem: In an Integral Domain R[x] If $a \in U(R) \Rightarrow ax+b$ ...
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0answers
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Prove that $\frac{Y}{X}$ is irreducible mod $Y^2-X^2(X+1)$

If we write $\frac{Y}{X} = \frac{F}{G}$, do we necessarily have $X$ divide $G$ and $Y$ divide $F$? If this can shed some light, I'm trying to find the set of poles of $\frac{Y}{X}$ mod $Y^2-X^2(X+1)$,...
2
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1answer
60 views

Abelian Galois group of even order

I'm stuck at the following problem. Let $a$ and $b$ be algebraic real numbers over $\mathbb{Q}$. Let $K= \mathbb{Q}(a+bi)$ be a simple extension of $\mathbb{Q}$. Suppose that $K$ is a Galois ...
1
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1answer
38 views

Irreducible fatorization of $X^n-1$ in both the ring of polynomials with complex coefficients and real coefficients

Let $f(x) = x^n-1$ be a polynomial in $\Bbb R[x]$. Factorize $f(x)$ as a product of irreducible polynomials in $\Bbb C[x]$ and show that if $n$ is even, $f(x)$ has two reals roots and if $n$ is odd, $...
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2answers
24 views

Number of elements in the ideal of Ring.

We have $$x^9+1 = (x+1)(x^2+x+1)(x^6+x^3+1)$$ is factorization of irreducible polynomials over $GF(2)$ (Galois field). Then we know that one of its ideal for the ring is $$R = GF(2)[x]/(x^9+1)$$ One ...
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1answer
14 views

Irreducibility of a polynomial over Rationals with condition given on its coefficients.

Let $f = a_nX^n+\cdots+a_1X\pm p \in \mathbb{Z}[X]$ with $\sum_{i=1}^n |a_i| < p$. Show that $f$ is irreducible in $\mathbb{Q}[X]$. Hint: Show that every root of $f \in \mathbb{C}$ has modulus ...
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2answers
30 views

Factorise the following p(x) as the product of a linear term and a quadratic polynomial with no real roots.

Factorise the following p(x) functions as the product of a linear term and a quadratic polynomial with no real roots (or if there are real roots, factorise to irreducible form and find them): 1. $$p(...
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4answers
35 views

Strategies for finding $[\mathbb{Q} (\sqrt{2} + \sqrt{3}) : \mathbb{Q} ]$

I need to find the degree of the extension $\mathbb{Q}(\sqrt{2} + \sqrt{3})$ over $\mathbb{Q}$. I don't quite know how to do it, nor can I exhibit any polynomial with root $\sqrt{2} + \sqrt{3}$, but I ...
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0answers
35 views

Notation and interpretation of Polynomials in $\mathbb{F}_{p}[x]$

i'm confused with some notation that involves reduction of polynomyals on $\mathbb{Z}[x]$ to $\mathbb{F}_p[x]$. It's part of the proof that Cyclotomic polynomials are irreducible over $\mathbb{Q}[x]$. ...
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0answers
31 views

Prove or disprove: $x^{p}+a$ is irreducible where $a\in \mathbb{Z}_{p}$.

Prove or disprove: $x^{p}+a$ is irreducible where $a\in \mathbb{Z}_{p}$. This is what I've done so far: By Fermat's Little Theorem, since $a^{p-1}\equiv 1$ (mod $p$) or $a^{p}\equiv a$ (mod $p$), ...
1
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1answer
76 views

Can this algorithm be fixed?

Consider the following algorithm from page 240 of this pdf: Irreducibility-Test(f) 1 $n ← \deg(f)$ 2 if $X^{p^n} \not\equiv X (\mod f)$ 3 $\quad$ then return "no" 4 for the prime divisors $...
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0answers
35 views

multiplication of polynomials in $\mathbb{F}_2[x]$

Let $p(x) = 1 + x + x^2$ and $q(x) = 1 + x + x^3$. Then is the multiplication $p(x)q(x)$ obtained like this: $$p(x)q(x)= (1 + x + x^2)(1 + x + x^3) = 1 +x +x^3 + x + x^2 + x^4 + x^2 + x^3 + x^5 $$ $$=...
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1answer
48 views

Why we ignore the other elements of $F_{p^k}$ when we check if any element of it is a root of multiplicity greater than $1$ in $p(x)$ or not?

$p(x) = x^4 + x +6$ . I was to find if there is any root of multiplicity greater than $1$ in the field of characteristics $p$. I was suggested to check every element of the $F_p$. But a filed ...
1
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1answer
50 views

How to prove that $x^5-5$ is irreducible over $\mathbb Q(\sqrt 2, \sqrt[3] 3) $

How to prove that $x^5-5$ is irreducible over $K = \mathbb Q(\sqrt 2, \sqrt[3] 3)? $ I came across this problem while solving another one and I dunno exactly how to proceed. I know by Eisenstein's ...
2
votes
1answer
45 views

If $\deg\left(f\right) = \min\left(\left\{d \in \mathbb{N}^\times;\; f\vert X^{q^d} - X\right\}\right)\,$, then $f$ is irreducible.

Let for some prime power $q$ $\mathbb{F}_q$ be a finite field and consider $f \in \mathbb{F}_q\left[X\right]$. I want to show the implication mentioned in the title, i.e. $$ \deg\left(\,f\right) = \...
0
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2answers
38 views

Irreducibility of $x^4 + x^3 + 1$ over finite field $\mathbb{F}_{2^{a}}$, $1 \leq a \leq 6$

I have to discuss the irreducibility of $P(X) = X^4 + X^3 + 1$ over finite field $\mathbb{F}_{2^{a}}$, $1 \leq a \leq 6$. So, for $a = 1$, we have that $P$ is irreducible since is has no roots in $\...
0
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2answers
39 views

Example of polynomial in two variables

Can you please give me an example of a polynomial $F \in K[X,Y]$ such that $V(F)$ is finite? I found in Fulton the following proposition: If F is an irreducible polynomial in $K[X,Y]$ such that $V(...
1
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1answer
60 views

Showing $x^4+x^2+x+1$ irreducible over $\mathbb{Z_3}$

$x^4+x^2+x+1$ irreducible over $\mathbb{Z_3}$. So since there are no roots there are no linear factors. From here do you just try to factor it as a product of 2 quadratics and show it that this leads ...
0
votes
1answer
33 views

Prove that $f$ or $g$ is reducible given that $\{rf+sg:r,s\in F[x]\} = N$ is a proper ideal and $\deg f \neq \deg g$.

Given a field $F$ and $f,g \in F$, I'm given the task of proving what it says in the title. So far, I've only managed to notice that $\deg f>0$ and $\deg g>0$, since if at least one were a '...
0
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2answers
32 views

Let $F$ be a field, when is the quotient ring $F[x]/(x^2+1)F[x]$ an integral domain?

Let $F$ be a field, when is the quotient ring $F[x]/(x^2+1)F[x]$ an integral domain? We know that for general rings, $R$ that $R/I$ is an integral domain if and only if $I$ is a prime ideal of $R$. ...
2
votes
1answer
43 views

$\gcd(a(X),\,a'(X))$ w.r.t. squarefree decomposition

We are considering polynomials over a field $\mathbb{F}$. For $a \in \mathbb{F}[X]$, we have a squarefree decomposition $$ a = \prod_{i=1}^k a_i^i $$ where $\gcd(a_i,\,a_j) = 1$ for $i \neq j$ and the ...
2
votes
2answers
44 views

Is the polynomial $x^2+1$ reducible in $\mathbb Z_7[x]$?

Is the polynomial $x^2+1$ reducible in $\mathbb Z_7[x]$? I think no, because $f(0)\ne0$; similarly, $f(1),f(2),f(3),f(4),f(5)$ and $f(6)\ne0$. Is it correct?
0
votes
1answer
45 views

$x^p - x+ a$ is irreducible in $\mathbb{F}_p [x]$ if $a\neq 0$ [duplicate]

I want to show that $f(x)=x^p - x+ a$ is irreducible in $\mathbb{F}_p [x]$ if $a \neq 0$. I know that if $b$ is a root of $f$, then $b+1$ is also a root of $f$. Can I use this fact to prove that $f$ ...