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Questions tagged [irreducible-polynomials]

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

3
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3answers
51 views

$p^{n-1}x^n - 1$ over $\mathbb{Q}$ for $p$ prime

Consider $f(x) = p^{n-1}x^n - 1 \in \mathbb{Q}[x]$. I want to show that it's irreducible when $p$ is prime. Neither reduction of the coefficients modulo some prime nor Eisenstein seems to work here. ...
1
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1answer
33 views

How to obtain the degree of this field extension?

Let $K/F$ be a field extension, $t,w \in K$ such that $t$ is transcendental over $F$ and $w$ is transcendental over $F(t)$. Prove that for any positive integer numbers $n$ and $m$ the equality $[F(t,w)...
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0answers
37 views

For which $f \in \mathbb{C}[x,y]$, $\mathbb{C}(x)[y]/(fy-\lambda)$ is isomorphic to $\mathbb{C}(t)$?

Let $f=f(x,y) \in \mathbb{C}[x,y]$ and denote $F_{\lambda}:=fy-\lambda$. Assume that for infinitely many $\mathbb{C} \ni \lambda$'s, $F_{\lambda}$ is irreducible in $\mathbb{C}[x,y]$, hence $(F_{\...
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0answers
29 views

For which $f \in \mathbb{C}[x,y]$, $fy-\lambda$ is irreducible in $\mathbb{C}[x,y]$ for infinitely many $\mathbb{C} \ni \lambda$'s

Let $f=f(x,y) \in \mathbb{C}[x,y]$, and for every $\lambda \in \mathbb{C}$, denote $F_{\lambda}:=fy-\lambda$. Is it possible to characterize all $f \in \mathbb{C}[x,y]$ such that $F_{\lambda}$ is ...
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0answers
22 views

For which $f \in \mathbb{C}[x,y]$, $fy$ is a field generator of $\mathbb{C}(x,y)$?

Let $f=f(x,y) \in \mathbb{C}[x,y]$. Call an element $F \in \mathbb{C}[x,y]$ a field generator of $\mathbb{C}(x,y)$, if there exists $G \in \mathbb{C}(x,y)$ such that $\mathbb{C}(F,G)=\mathbb{C}(x,y)$....
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0answers
116 views

Verify that if $u$ is irreducible in $Z_n$ , then $u$ is irreducible in $Z_n[x]$. [on hold]

This looks so simple but .. I can't prove or disprove this. [Definition (irreducible element of ring)] ☞ Let $R$ is commutative ring with unity $1$. $u$ is irreducible elements in $R$ iff $u$ is ...
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1answer
57 views

Show that $\mathbb{Z}_3[i]$ is a field

I know that $\dfrac{\mathbb{Z}_3[x]}{\langle x^2+1\rangle}$ is isomorphic to $\mathbb{Z}_3[i]$, does this help me prove that $\mathbb{Z}_3[i]$ is a field? $\langle x^2+1\rangle$ is the ideal ...
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2answers
41 views

Check the statements about irreducibility

We have the ring $R$ and the polynomial $a=x^3+x+1$ in $R[x]$. I want to check the following statements: If $R=\mathbb{R}$ then $a$ is irreducible in $\mathbb{R}[x]$. This statement is false, ...
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4answers
63 views

Splitting field of $x^{21}-1$ over $\mathbb F_3$

Let $\mathbb F_3$ be the field with 3 elements and $\overline{\mathbb F}_3 $ its algebraic closure. Let $K$ be the splitting field of $f(x)=x^{21}-1$ (I guess over $\mathbb F_3$). Find the number of ...
2
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1answer
31 views

counting Misiurewicz points

I enumerated the number of Misiurewicz points using SageMath to factor into irreducible polynomials over $\mathbb{Z}$, where the degree (after discarding factors corresponding to lower (pre)periods) ...
7
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4answers
133 views

$x^4+x^3+x^2+x+1$ irreducible over $\mathbb F_7$

This question came from the answer here. The answer claims that $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb F_7$. I can check that it has no roots in the field, but why can't it be written as a ...
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1answer
43 views

Why or why not use an irreducible polynomial for a cyclic redundancy check?

I understand the need for using an irreducible polynomial for a prime power finite field when doing multiplication with numbers in those fields. For certain applications, such as the Q parity bytes ...
3
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1answer
57 views

Minimal polynomials of elements in $\mathbb F_{16}$

List all polynomials in $\mathbb F_2[x]$ that are minimal polynomials of elements from $\mathbb F_{16}$. Since minimal polynomials are irreducible, this problem just asks to list irreducible ...
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4answers
148 views

Prove that the polynomial $x^5+2x+1$ is irreducible over $\mathbb{Z}$.

Show that there do not exist polynomials $p(x)$ and $q(x)$ , each having integer coefficients and of degree greater than or equal to 1 such that: $$p(x)\cdot q(x) = x^5 + 2x + 1 $$ My approach I ...
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2answers
176 views

What are the steps involved in finding the Greatest Common Divisor of two polynomials?

Ultimately I'm trying to define all the steps necessary to go from this toy quartic polynomial modulus: $$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$ to: $$x = 18, 19$$ One of the recommended ...
3
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2answers
67 views

Divisibility of polynomials

I am having trouble trying to prove the following statement: Let $f = (f,f').g$ with $(f,f')\not= 1$. Both $f$ and $g$ are polynomials with rational coefficients. Then, $(f,f')$ and $g$ share roots. ...
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1answer
35 views

Splitting field of polynomial is F8

I've got a comprehension question: Be the polynomial: $f(x) = x^3 + x + 1$ over $\mathbb{F}_2[X]$ I know, that it's splitting field is $\mathbb{F}_8$, but that means, $f$ splits into linear factors ...
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2answers
151 views

Irreducibility test in a number field

While thinking about this question I noticed that if $\alpha$ is a root of $f(x) = x^9 + 3 x^6 + 165 x^3 + 1$ then $$\left(\frac{\alpha^3+1}{3 \alpha}\right)^3 = -6$$ so $\mathbb{Q}(\alpha)$ contains ...
5
votes
3answers
200 views

Irreducibility of higher order polynomials over $\mathbb{Q}$

I wish to show that the polynomial $f(x)=x^{9}+3x^{6}+165x^{3}+1\in\mathbb{Q}[x]$ is irreducible over $\mathbb{Q}$. My guess; reducing $f(x)\in\mathbb{F}_{p}[x]$ for suitable prime $p$, it might be ...
0
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1answer
27 views

if $p(x)$ is irreducible show $cp(x)$ also is irreducible

If $p(x)$ is irreducible in $K[x]$ and $c \ne 0$, $c \in K$ $\Rightarrow cp(x)$ is irreducible in $K[x]$. I would start saying that $p(x)=a(x)b(x)$ with $a(x)$ a unit $\lambda \in K$, without loss ...
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0answers
52 views

Polynomial of degree $5$ reducible over $\mathbb Q(\sqrt 2)$

Give an example of an irreducible monic polynomials of degree (a) $4$; (b) $5$ in $\mathbb Z[x]$ that is reducible over $\mathbb Q(\sqrt 2)$, or prove that none exists. I managed to find a polynomial ...
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0answers
45 views

Test of irreducibility of a binary polynomial

What criteria do we have to efficiently test if a binary polynomial $P(x)$ is irreducible? Assume $P$ is given as a vector of $n+1$ bits $c_i$, with $c_0=c_n=1$, as $P(x)=\displaystyle\sum_{j=0}^nc_j\...
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votes
1answer
28 views

Show that there is a polynomial such that P(n) is not prime [closed]

Let m and n be a integer. Show that for all values of n there is a polynomial such that P(n) equals toma prime number. For instance for the polynomial $$x^{2}+1$$ for x=1 the result is equal to 2. ...
1
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1answer
36 views

Show that prime polynomial is irreducablr [closed]

Let p be a prime number. Question is for any prime value of p the polynomial $$1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...++\frac{x^p}{p!}$$ is irreducable. I found out that it seems like a ...
2
votes
2answers
42 views

Can it happen that an irreducible degree $2$ polynomial becomes a perfect square in the algebraic closure?

Question. Is it possible to find a field $K$ together with a degree-$2$ polynomial $P \in K[x]$ such that $P$ is irreducible, but $P = Q^2$ for some $Q \in \overline{K}[x]$? This can't occur if $K = \...
0
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1answer
61 views

Minimal polynomial for $\sqrt[5]{2}$ over $\mathbb Q(\sqrt[]{3})$

Find the minimal polynomial for $\sqrt[5]{2}$ over $\mathbb Q(\sqrt[]{3})$. The natural candidate is $x^5-2$. But the problem is to prove it is irreducible. In theory, I can write down the roots ...
4
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0answers
19 views

Period of $i\mapsto S_0\,x^i\bmod P$

Let $P$ be a given polynomial of degree $n$ with coefficients in a finite field. Let $S_0$ be a given polynomial of degree less than $n$ with coefficients in that field. How do we derive the (...
0
votes
1answer
38 views

Prove the following lemma(dealing with polynomials)

If $p$ is a prime number and $ a_0,a_1,\ldots , a_{p - 1} $ are rational numbers satisfying $$ a_0 + a_1 \alpha + a_2 \alpha ^2 + \ldots + a_{ p - 1} \alpha ^{ p - 1} = 0 $$ where $$ \alpha = \cos(\...
1
vote
2answers
33 views

elementary polynomial divisibility question

I was reading Dummit and Foote, and in there, there is a line that goes like this: Since for prime $p$, $p^2-1$ is divisible by 8, $x^{p^2-1}-1$ is divisible by $x^8-1$. (over $\mathbb{F}_{p}$) ...
3
votes
1answer
68 views

Set of primes $p$ for which $x^3-4x-1$ factors completely over $GF(p)$

Find the set of primes $p$ (either a modular congruence or quadratic form) such that $x^3-4x-1$ factors into three linear factors over the field $GF(p)$. I noted that these primes $p$ include $37, 53, ...
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3answers
54 views

Condition for having order 2 elements in Galois group for polynomials over $\mathbb{Q}$

Suppose we have $f \in \mathbb{Q}[X]$, with only real roots. Then the complex conjugation is not an automorphism, but is this enough to say that there exist no order two elements in $\text{Gal}(f)$? ...
3
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1answer
87 views

Factorize $x^8-x$ over $F_3$ and $F_{81}$

Consider the polynomial $p(x)=x^8-x$ in $F_3$: (a) find the splitting field of $p$ over $F_3$ and factorize $p$ over $F_3$ (b) factorize p over $F_{81}$ (a) If the roots of $p$ are all ...
5
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0answers
96 views

Fastest way to factorize $x^{18}-x^3$ over $\mathbb{F}_2$

$x^{18}-x^3=x^3(x^{15}+1)=x^3(x^{5\cdot3}+1)=x^3(x^5+1)(x^{10}+x^5+1)$ $x^5+1$ has the root $1$ in $\mathbb{F}_2$ so using the factor theorem I got: $x^5+1=(x+1)(x^4+x^3+x^2+x+1)$ (how to prove that ...
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2answers
88 views

Proving that the polynomial $10X^6-15X^2+7$ is irreducible in $\mathbb{Q}[X].$

As stated, I am trying to prove that $10X^6-15X^2+7$ is irreducible in $\mathbb{Q}[X].$ I have been given the hint to compare the above polynomial with $7X^6 - 15X^4+10.$ I know that $7X^6 - 15X^4+...
2
votes
1answer
33 views

Irreducibility of a polynom [duplicate]

Any thoughts on how to show that $$p(x)=-1+\prod_{i=1}^n (x-i)$$ is irreducibile in $\mathbb{Z}[x]$?
2
votes
1answer
36 views

Irreductibility of a polynom

I have to show that $p(x)=x^5+5x+11$ is irreductible on $\mathbb{Z}[x]$ both using and not using Eisenstein. I have been trying to translate $p(x)$ or searching a field $\mathbb{Z}p$ but can't reach ...
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1answer
47 views

How can I show that irreducibility for this set?

$k$ is an algebraically closed field. $X$ $\subseteq$ $A^n$ is an affine irreducible variety and $f \in k[x_1,...,x_n]$ $X_f = \{ (x_1,...,x_n) \in X \mid f(x_1,...,x_n) \neq 0\}$ $Y:=\{(x_1,...,x_n,...
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1answer
26 views

Homomorphism and irreducible polynomials

Let R[x] be a polynomial ring. Let S[x] be another polynomial ring such that $R[x]\subset S[x]$ . Let $\phi: R[x]\rightarrow S[x]$ be an inclusion homomorphism. Let $f(x)$ be a polynomial in $R[x]$ ...
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1answer
49 views

If the homomorphic image of a polynomial is irreducible, does it imply polynomial is irreducible [closed]

If the homomorphic image of a polynomial is irreducible, then is the original polynomial irreducible? let $\phi: R[x]\rightarrow S[x]$ be a ring homomorphism. Let $f(x)\in R[x]$ be a polynomial and $...
5
votes
3answers
67 views

Is the ideal $I:=\langle xy-z,x^5-z^3 \rangle $ prime in $\Bbb{C}[x,y,z]$?

Let us take the ideal $I:= \langle xy-z,x^5-z^3 \rangle $ of the ring $\Bbb{C}[x,y,z].$ We want to find if this ideal is prime. My thoughts: We define $f:=xy-z,\ g:=x^5-z^3 \in \Bbb{C}[x,y,z].$ The ...
1
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1answer
49 views

Least degree polynomial with integer coefficient and having one root $\sqrt{8+\sqrt 6 +4\sqrt 3+ 3\sqrt 2}$

Find Least degree polynomial with integer coefficient and having one root as $\sqrt{8+\sqrt 6 +4\sqrt 3+ 3\sqrt 2}$ I tried doing square again and again is there any simpler method
0
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1answer
41 views

Find all the monic irreducible polynomials of degree less than equal to $3$ in $F_2[X]$, and the same in $F_3[x]$.

Find all the monic irreducible polynomials of degree less than equal to $3$ in $F_2[X]$, and the same in $F_3[x]$. My Attempt: We need to have $1$ in the expression of our polynomial since ...
0
votes
2answers
31 views

Irreducible polynomial is square-free modulo almost every prime

Let $f \in \Bbb Z[X]$ be irreducible. What is the easiest way to show that there are only finitely many primes $p$ such that $f \pmod p$ is not square-free (i.e. not separable) in $\Bbb F_p[X]$ ? Is ...
11
votes
1answer
203 views

What are the factors of this quotient given by Fermat's Little Theorem?

$\forall a,b \in \mathbb{Z}, p\in \mathbb{P}$, let $$F_p(a,b) = \frac{(a+b)^p-a^p-b^p}{p}$$ Note: $F_3 = ab(a+b)$ $F_5 = ab(a+b)(a^2+ab+b^2)$ $F_7 = ab(a+b)(a^2+ab+b^2)^2$ According to data ...
2
votes
1answer
55 views

For $p$ prime, is the polynomial $x^p-x+1$ irreducible in $\mathbb{Z}_p$? [duplicate]

It is possible, for $p\in\mathbb{N}$ prime, that the polynomial $x^p-x+1$ is irreducible in $\mathbb{Z}_p$? By the identity $a^p\equiv a$ mod $p$ for any $a\in \mathbb{Z}_p$ surely there is not a ...
1
vote
1answer
68 views

How to check if $x^{21} + 2x^8 + 1$ and $x^{21} + 2x^9 + 1$ are irreducible in $\mathbb{Z}_3$?

The full question is to answer if these two polynomials have multiple zeros. There is a theorem in my book that says that if $f(x) \in F[x]$ and $\operatorname{char}(F) = p ≠ 0$ then $f(x)$ has ...
5
votes
1answer
223 views

Does every polynomial over a finite field have a square root modulo an irreducible polynomial?

Given a polynomial $p \in \operatorname{GF}(2^m)[x]$ and an irreducible polynomial $g \in GF(2^m)[x]$, is there a $d \in \operatorname{GF}(2^m)[x]$ such that $d^2(x) = p \pmod{g(x)}$? In other words, ...
0
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1answer
16 views

Prime Polynomial and roots

We have a sentence about prime polynomials over a field $K$. It's stated that whenever a polynomial $f(x) \in K[x]$ with $deg f(x) \ge 2$ has a root $c$ in $K$, it's clear that $(x-c) \mid f(x)$ and ...
1
vote
1answer
59 views

A field with an irreducible, separable polynomial with roots $\alpha$ and $\alpha + 1$ must have positive characteristic.

Given a field $\mathbb{F}$ with an irreducible, separable polynomial $f(x),$ let $E$ denote the splitting field of $f$ over $\mathbb{F},$ and assume that $\alpha$ and $\alpha + 1$ are roots of $f(x).$ ...
1
vote
3answers
52 views

Polynomial decomposition into irreducible factors

Decompose $x^5 + x + 1$ into irreducible factors in $\mathbb{Z}_2[x]$. I would like to know how to reason and how to proceed. I am a beginner in this field of mathematics, and I am trying to ...