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

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2
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46 views

How to factor intricate polynomial.

I would like to know how to factor the following polynomial. \begin{equation} ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c \end{equation} What is the method i should use to factor it? If anyone could help.. ...
3
votes
1answer
44 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 ...
1
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0answers
33 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 ...
1
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2answers
58 views

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 ...
4
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1answer
39 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 ...
0
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1answer
43 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
60 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
21 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 $Z[x]$?
2
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0answers
16 views

$a$ 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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2answers
40 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 ...
0
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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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0answers
31 views

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: ...
2
votes
1answer
60 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 ...
3
votes
1answer
49 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 $c^{d+1} \gt a^b \ge c^d$ ...
4
votes
1answer
143 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 ...
2
votes
1answer
28 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 ...
8
votes
2answers
64 views

Irreducibility of Cyclotomic polynomials over number field

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 most ...
1
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1answer
35 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
21 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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2answers
39 views

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 $? ...
3
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0answers
18 views

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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3answers
137 views

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 ...
0
votes
1answer
14 views

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 ...
2
votes
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 ...
1
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1answer
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
19 views

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 ...
-1
votes
3answers
83 views

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
43 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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0answers
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
40 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 ...
3
votes
1answer
49 views

finding all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$, faster method?

I want to find all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$ and I wonder if there's a better method than the following. The polynomials are of form $aX^2 + bX +c$. So I have to ...
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2answers
25 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
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2answers
26 views

Constructibility of roots of a polynomial

I`m trying to decide if the roots of the polynomial $f(x) = x^4+x^3-2x^2 +x +1$ is constructible. My first thought was to show that the polynomial f is irreducible in $\mathbb{Q}$ then for any root ...
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3answers
73 views

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ [duplicate]

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ Attempt. Since $x^2+x+1$ is of degree 2, it is reducible iff it has a root in $\mathbb{F}_p$. It has a root in ...
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2answers
48 views

Cases where irreducibility of polynomials over $\mathbb F \iff$ it has no roots in $\mathbb F $

This is question is in light of comments in this question and another question I asked few days back. Quoting Hayden from the first link, You can use Rational Root Theorem to show a polynomial is ...
2
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1answer
54 views

Irreducibility of polynomials $x^{2^{n}}+1$

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some "easy" proof for that (where easy means not using a big theory like ...
0
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1answer
18 views

multiple roots of irreducible polynomial 2

let say we have an irreducible polynomial over field $F$. I need to prove that all roots of f have the same multiplicity. I know that if $\text{Ch}(F)=0$ so this is easy but I don't know what to do ...
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1answer
33 views

Solve $\int (y^4+by^2+c)^{-1/2}dy=x$ as y is a real valued function of x.

Solve $\int (y^4+by^2+c)^{-1/2}dy=x$ as y is a real valued function of x. Here $b$ and $c$ are real constants so that $c>b^2/4$.
5
votes
3answers
123 views

Showing that $x^4 -2x^2 +8 x+1$ is irreducible over $\Bbb Q$

I want to show that the polynomial $$f(x)= x^4 -2x^2 +8 x+1$$ is irreducible over $\Bbb Q$. I've proved it by a long method, but I need an easy and short method. I've try to put $x=t+1$, but this ...
0
votes
1answer
38 views

Irreducibility of a cubic polynomial

Let $f(x)=x^3+2x^2+x-1$. Then over which of the following fields $k$ is $f$ irreducible? $k=\mathbb{Q}$ $k=\mathbb{R}$ $k=\mathbb{F}_2$ $k=\mathbb{F}_3$ My Attempt: (2) $f$ is ...
8
votes
1answer
131 views

Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
5
votes
1answer
73 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
6
votes
6answers
139 views

Show that $x^{3}-3$ irreducible over $\mathbb{Q}(\sqrt{-3})$

Is there a slick way to show that $x^{3}-3$ is irreducible over $F= \mathbb{Q}(\sqrt{-3})$? What I did seems kind of convoluted (showing directly that there is no root in F). Thanks
3
votes
2answers
29 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
0
votes
0answers
30 views

factorization of a cubic polynomial in Z_(p)

here is my exercise (this is training before the exam in a few weeks...): "Consider the polynomial $P(X)=X^3-X^2-1$. What is the factorization of $P$ in $\mathbb{Z}_{p}[X]$?" What I have done so far ...
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2answers
87 views

Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
1
vote
2answers
144 views

How to find the irreducible polynomial?

It is giving me a lot of trouble, and I'm beginning to think it's not possible. Find $\operatorname{irr}(2\sqrt{2} + \sqrt{7})$. I start like this: $x = 2\sqrt{2} + \sqrt{7}$ I have squared ...
4
votes
1answer
84 views

Irreducibility of $X(X-3)(X-\alpha)(X-\beta) + 1$

I'm trying to solve the following exercise: Show that for $\alpha,\beta\geq 3$, the polynomial $f = X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check ...