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

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Can Two Different Polynomials Agree on an open interval? [duplicate]

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)? TBN: This question is partially answered in ...
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1answer
57 views

Overlapping Polynomials

This question is related to this:Interpolating Polynomial & It's Root We have $P_3=P_2\cdot P_1$,for three non-zero polynomials. The degree of each polynomial is at least 1. Question: Does ...
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Irreducibility of $p(x)$ implies that of $p(x+c)$ only when taken over a field?

$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now ...
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1answer
30 views

Irreducibility of a polynomial

Given that $\mathbb F$ is a field and $\mathbb F[x]$ is the polynomial ring over $\mathbb F$. $\ \ $If the polynomial $a_{0}+a_{1}x+a_{2}x^{2}+......a_{n}x^{n}$ is irreducible over ...
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9 views

Probability That a Polynomial has Specific Root when we use Permutation Polynomial

To some extent similar question was asked here: Polynomial Interpolation and Security Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-2$, ...
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1answer
47 views

minimal polynamial of $i\sqrt{5}+\sqrt{2} \in C$

Find the minimal polynamial of $i\sqrt{5}+\sqrt{2}\in C$ over rational numbers. My solution is; Say $\alpha=i\sqrt{5}+\sqrt{2}$ $(\alpha-\sqrt{2})^2=(i\sqrt{5})^2$ $\alpha^2-2\sqrt{2}\alpha=-5$ ...
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4answers
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How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials?

Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: ...
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1answer
27 views

Irreducible inseparable polynomial of prime degree has derivative zero

Let $F$ be a field and let $f(x)$ be irreducible polynomial of degree $p$ "prime" in $F[x]$ having splitting field $K$. Let $[K,F]=pt$, for some integer $t$, then: 1) $f(x)$ is irreducible over $F$ ...
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Irreducibility Implication between $\mathbb{Q}[x]$, $\mathbb{Z}[x]$, and $\mathbb{Z_p}[x]$ [closed]

These following questions might be pretty easy, but I am really confused about them. Let $f(x)$ $\in \mathbb{Z}[x]$, $p=$prime. 1- Does irreducibility of $f(x)$ in $\mathbb{Z_p}[x]$ imply ...
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28 views

Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive

Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive. Prove also that even though the Galois group of $f$ is transitive not every permutation ...
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1answer
32 views

Finding the Irreducible polynomials which are primitive as part of Coding theory course

I am taking a coding theory course where we have to be able to work out which polynomials over a field $\mathbb F_q$ are irreducible or reducible and then which of the irreducible ones are primitive. ...
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2answers
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Irreducibility criteria for polynomials with several variables.

Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime. If it is $K[x]$, then there are several methods which can be used to check whether a given ...
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1answer
62 views

Show that $1-i$ is irreducible in $\mathbb{Z}[i]$? [closed]

I am using contemporary abstract algebra, by Joseph, I am in divisibility factor, can you please provide details step to solve above example with explanations.
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Determining the minimal polynomial over $\Bbb{Q}$

I was working on a homework assignment from Hungerford: Find the minimal polynomial of the element $\sqrt{1+\sqrt{5}}$ over $\Bbb{Q}$. Naturally the solution would be the polynomial with roots ...
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1answer
45 views

Find out whether a polynomial is irreducible or not

Let $f=X^7-(7-6i)X^3+5X^2+3+6i\in\mathbb{Z}[i][X]$. Check whether $f$ is irreducible: over $\mathbb{Z}[i]$ over $\mathbb{Q}(i)$ Probably I will have to use Einstein criterion with some ...
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1answer
35 views

Showing irreducibility of a polynomial. [duplicate]

How would you go about showing that $p(x)=\frac{x^5-1}{x-1}=x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{Q}$. I'm having trouble seeing how one can show whether this kind of polynomials are ...
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1answer
41 views

Gauss' Lemma and repeated factors

I am reading a proof showing that $x^5+y^7 +z^{11}$ is irreducible in $\mathbb{C}[x,y,z]$. We use the natural isomorphism to show that $\mathbb{C}[x,y,z] \cong \mathbb{C}[y,z][x]$. We want to use ...
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0answers
16 views

The number of monic irreducible

I have some questions about the number of monic irreducible polynomial. Let $a_n$ be the number of monic irreducible polynomial of degree $n$ over finite field of $q$ elements $F_q[X]$. Then we ...
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3answers
50 views

How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $

I would like to know how to factor the following polynomial. $$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$ What is the method i should use to factor it? If anyone could help.. Thanks in advance.
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1answer
49 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...
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0answers
34 views

$x^p -x-1$ irreducible over $\mathbb{F}_{p}$ [duplicate]

Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$. I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is ...
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2answers
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Problem on Galois theory and irreducible polynomial

Let $p,q$ be primes, estimate the degree $[\Bbb Q(\sqrt[p]{2}\cdot\sqrt[q]{2}):\Bbb Q]$ and prove that the polynomial $X^q-2$ is irreducible in the ring $\Bbb Q(\sqrt[p]{2})[X]$ I found this problem ...
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1answer
40 views

Irreducible polynomials with one large coefficient

Is it true that for every monic polynomial $p(x) \in \mathbb{Z}[x]$, $p(0)\neq 0$, of degree $n>0$ there exists a real number $M>0$ such that for every $|m|>M$ and for every $k$ odd integer ...
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1answer
45 views

Separable polynomials on field with char 2

On a field $K$ with $char(K)$ not equal to 2, all irreducible polynomials of a quadratic extension are separable. The proof is straightforward: Assume the opposite, namely $P=X^2+aX+b = ...
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4answers
61 views

Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$.

Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$. In the text, we haven't gotten to the theorem that the roots of polynomials are the only factors , and I would rather not prove it in this ...
2
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1answer
22 views

If $f,g$ in $Z[x]$, $h$ in $R[x]$ with $f=gh$, is $h$ nessecarily in $Z[x]$?

Let $f$ and $g$ be monic polynomials in $Z[x]$. There exists a polynomial $h$ in $R[x]$ such that $f=gh$ for all real $x$. Is $h$ nessecarily in $Z[x]$?
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$a$ and $b$ irreducible polynomials such that $\forall u \in \mathbb{Q}[t], a|u(t^n)\iff b|u(t^n)$

A little context is in order. I was trying to find counter-examples to the following statement: $$\phi : X\rightarrow Y \;\text{injective} \Rightarrow \phi\otimes K : X\otimes_k K \rightarrow ...
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42 views

Let $K=GF(2)$ and $p(x)= x^3 + x+1.$ Show that $p$ is irreducible in $K[x]$

Let $K=GF(2)$ and $p(x)= x^3 + x+1$ Show that $p$ is irreducible in $K[x]$ First of all am I right in interpreting: $$GF(2) = \mathbb Z / 2 \mathbb Z= \{ 0,1\}$$ So basically, $p(x)$ is a ...
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1answer
27 views

Why doesn't $f_1, f_2$ have an inverse polynomial?

Consider $f\in R=F[X]$. It is given that $f$ doesn't have an inverse but it's reducible. Therefore, there are $f_1,f_2$ such that $f=f_1f_2$, where $f_1, f_2$ also doesn't have an inverse polynomial. ...
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Why is there a $q_i$ such that $q_j|q_i$?

Let $q_i$, a sequence of of irreducible polynomials where $q_i$'s highest-order term has coefficient $c_n = 1$ (by the way, what's the right term to describe this property?) Anyhow, let's look at: ...
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1answer
61 views

All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
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1answer
51 views

Easy difference of exponents ($a^b$ - $c^d$) for arbitrarily large numbers

I am wondering if there is an easy way to calculate the difference of two exponents, with different bases, without calculating the number. If I have $a^b$ - $c^d$, where $c^{d+1} \gt a^b \ge c^d$ ...
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1answer
159 views

Nature of roots of a biquadratic equation

(Biquadratic $\rightarrow$ Quartic (degree 4)) The Question: (from a book i am practicing from) Find the nature of the roots of the equation $$f(x) = 45 x^4-144 x^3+146 x^2-56 x+12=0$$ (By nature i ...
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1answer
29 views

Count the number of monic irreducible polynomials of degree 12 over $\mathbb F_q$

This is a qualifying problem. I cannot understand how the inclusion exclusion principle work here in detail. However, I have an argument which leads to a different answer. I am not sure ...
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Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
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1answer
38 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
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2answers
22 views

Characteristic is positive and exist polynomial with $g(x^p)=f$

$F$ is a field. $f \in F[X]$ is inseparable and irreducible. Show that characteristic p of F is positive and there exists $g$ with $g(x^p)=f$. We know that f is inseparable so $gcd(f,f')\neq 1$, so ...
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Can two irreducible polynomials have different powers of the same real number as roots?

Say we have two irreducible polynomials in $Q [x] $. We call them $f, g$. Say one of the roots of $f $ is $a$. Is it possible that $g$ satisfies a root of the form $a^n$ for some natural number $n $? ...
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For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
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How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
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1answer
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module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
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2answers
43 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
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58 views

Showing a polynomial is irreducible over $\mathbb{C}[x,y]$

Given $m,n \in \mathbb{N},$ how can I show that the polynomial $x^m+y^n-1$ is irreducible in $\mathbb C[x,y]$? I'm given the following hint, but I don't follow. Note: I know Eisenstein's ...
2
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1answer
43 views

Irreducible polynomial modulo 2

I need to prove that polynomial $f(x) = x^{10}+x^{3}+1$ is irreducible modulo $2$. It is irreducible if $f|x^{1024}-x$, isn't it? I can use polynomial long division to check it, but this is not ...
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1answer
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Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
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3answers
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Factor the polynomial [duplicate]

Factor the polynomial $X^3-X+1$ in $F_{23}$ and $X^3+X+1$ in $F_{31}$. How can I know in which way to factor a polynomial mod $p$? Is there some specified method to do that? Thanks.
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4answers
71 views

Expansion of $x^n-y^n$

Studying polynomials I couldn't find a way to expand $x^n-y^n$ as a product of other polynomials. Now of course we know that $$x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)$$ and I came up with this: ...
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1answer
44 views

Prove the polynomial is irreducible [duplicate]

I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1. p is a prime ...
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30 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
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1answer
42 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...