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Questions tagged [irreducible-polynomials]

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

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183 views

Irreducibility in a polynomial related to quadratic residues

From Romania TST 2004 Day 5 P3, I was introduced to the polynomial $$f(x)=\sum_{i=1}^{p-1} \left( \frac{i}{p} \right)x^{i-1}$$ This polynomial is clearly not irreducible - $x=1$ is a root. Even more ...
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Can two monic irreducible polynomials over $\mathbb{Z}$, of coprime degrees, have the same splitting field?

Let $f,g \in \mathbb{Z}[X]$ be monic polynomials. It is possible for distinct monic polynomials over $\mathbb{Z}$ to have the same splitting field. For example $f = x^4 - 2$ and $g= x^4+2$ both have ...
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63 views

Irreducibility of $P(X)-P(Y)+1$

Let $P\in{\mathbb Q}[X]$ be a non-constant polynomial. Consider the polynomial $Q(X,Y)=P(X)-P(Y)+1$. Is $Q$ always irreducible in ${\mathbb Q}[X,Y]$ ? Is $Q(X,y)$ always irreducible in ${\mathbb Q}[...
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110 views

If prime $p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0$ then $f(x)=a_nx^n+\ldots+a_0$ is irreducible in $\mathbb{Z}[x]$

I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it: If we express a prime $p$ in base $10$ as $$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,...
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191 views

Irreducibility of a family of polynomials coming from Fibonacci polynomials

Let $\{F_n(z)\,|\, n\geq 1\}$ be the Fibonacci polynomials, defined recursively by $F_1=1, F_2=z$ and $F_{n+2}=zF_{n+1}+F_n$. Now consider the polynomial $$\varphi_{n,m}(z)=4-F_n^2F_m^2(z^2+3).$$ I ...
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284 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If $...
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91 views

Irreducibility of q-factorial plus 1

Is it true that $[n]_q! + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$ ? I checked that this is true for $n$ up to $20$. Here $[n]_q! := 1 (1 + q) (1 + q + q^2) \...
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62 views

Probability that an irreducible polynomial has a root modulo a prime $p$

Let $q$ be an irreducible polynomial over $\mathbb{Z}.$ What is the probability that $q$ has at least one root modulo a prime $p?$ For quadratic $q,$ the probability should be about a half by ...
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117 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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For what irreducible $f\in\mathbb{Q}[X]$ is this sequence periodic modulo $f$?

This is just an interesting problem I found in an old notebook of mine: Given an irreducible $f\in\mathbb{Q}[X]$, is there a way to determine whether the sequence $(p_n)_{n=0}^{\infty}\subset \...
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188 views

Irreducible polynomial such that its roots satisfy a given relation $x_3 = f(x_1,x_2)$

My general question is: What are the polynomials $f \in \Bbb Q[X,Y]$ such that there exists an irreducible polynomial $P \in \Bbb Q[X]$ having three distinct zeros $x_1,x_2,x_3$ that satisfy $x_3 = ...
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208 views

Eisenstein's criterion for two variables

I want to know if there is a criterion, like Eisenstein's criterion, for polynomials with 2 variables? If there is, how does it work?
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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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Why Fibonacci LFSR random number generation works?

If I use primitive polynomial of GF($2^m$) in Fibonacci LFSR, it is generating all m-length binary combinations. But, I cannot understand why this should happen. I am not getting any mathematical ...
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204 views

Irreducibility of polynomials via Frobenius map

I am having trouble trying to show this: Let $f \in \mathbb{F}_p[x]$ be a non-constant polynomial and let $F$ denote the Frobenius map $F: R \rightarrow R$ where $R = \mathbb F_p[x]/(f)$. Prove ...
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24 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 (...
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105 views

A rational function with hidden symmetry and alternating poles and zeros.

Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties. Consider the function: $$ f(m,n_1,n_2;z)=\frac{1}{...
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193 views

Primitive element and choice of irreducible polynomial

It is known that every finite field of the same order $p^k$ are isomorphic. So, $F_p[x]/\langle q(x)\rangle$ leads to the same field for any choice of irriducible k-degree $q(x)$ over $F_p[x]$. But, ...
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Irreducibility in $\mathbb{F_2}[X]$ and field extensions

Here's an exercise I found: 1) Show that $$f(X):=X^3+X+1$$ $$g(X):=X^3+X^2+1 $$ are irreducible in $\mathbb{F_2}[X]$. 2) Let $a, b \in \overline{\mathbb{F_2}}$ with $f(a)=g(b)=0$. $$\mathbb{...
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597 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
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171 views

Linear independence of roots

Given an irreducible polynomial $P(x)\in K[x]$ where $K$ is a field, what are the criteria for the roots of $P$ to be linearly independent over $K$? Edit: fixed in response to comments below
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172 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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94 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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118 views

Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see <...
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1k views

Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
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Maximum possible number of extrema of the function?

Consider a function : $$ f(x)= P(x)e^{-(x^4+2x^2)} $$in the domain $x \in (-\infty,\infty)$, $P(x)$ is any polynomial of degree $k$. What is the maximum possible number of extrema of the function. ...
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83 views

Question about Lang's Chapter 6 Theorem 9.1

I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields. He begins his argument with a lemma, If $q \equiv 1$ mod $l$ for ...
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35 views

How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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determining if quotient ring of polynomials over a finite field is a field or not

I am stuck with this question: "Determine if $GF(2011^2) [x] /<x^4-6x+12>$ is a field or not." I know that since this polynomial ring is defined over a field, I only have to determine if $x^4-...
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How to prove $x^5 + y^7 +11$ irreducible in $\mathbb{Z}[x,y]$

I’m so wired attacking reducibility using the fact that I’m on a field and playing with orders of elements to show there is no linear or quadratic factors. But $\mathbb{Z}$ is not a field and the ...
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Which field has the property that there is an irreducible polynomial of any degree over it?

I have known that there is an irreducible polynomial for any degree over rational field. However,for other fields such as finite field and extension fields for the rational field,what is the ...
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80 views

Optimal normal basis in Tower field construction

If I work over $GF(2^4)$, then the condition for the type I optimal normal basis holds, in fact $4+1=5$ is prime and $2$ is a primitive element in $\mathbb{Z}_5$. My question is about the existence of ...
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33 views

Prove that $ f(X)=X^{p^{n}}+p-1 $ is irreducible in $ \Bbb{Z}[X] $- a reduction approach?

Here $ n,p $ are positive integers with $ p $ being a prime number. I proved the result below, but I would like to know if there is an alternative way of doing it using reduction modulo $ p $. Note ...
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235 views

Proof check: Irreducibility of $xy-1$

Mainly I ask, to check if there are holes in my logic anywhere locally, since these will affect me globally, and perhaps other readers. If I want to prove that $xy-1\in \Bbb{C}[x,y]$ is irreducible, ...
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155 views

Factor the two polynomials into a product of irreducible elements of $\mathbb{Q}[x]$

I need to find a factorization of both $f_{1}=2x^{2}+4x+6$ and $f_{2}=2x^{2}+4x-6$ into a product of irreducible elements of $\mathbb{Q}[x]$. I already was able to do so in the case of $\mathbb{Z}[x]$...
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131 views

Which of the following ideals is maximal in $\mathbb{Z_{3}}[x]$

I need to determine which, if any, of the following ideals in $\mathbb{Z}_{3}[x]$are maximal: $$\mathbf{(2)},\, \mathbf{(x+1)},\, \text{and}\, \mathbf{(x^{2}+x+1)}$$ (i.e., the principal ideals ...
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61 views

How many roots does one add when considering the quotient of a polynomial ring?

The standard method (I think) for proving that decomposition fields of polynomials exist is by considering an irreducible polynomial $P$ (over the field $K$), and then noticing that $X + (P)$ is a ...
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181 views

Understanding symbol/notation in Abstract Algebra

I know that: $F(\alpha) = F[X]/f(x)$ where $F[x] = \{a_{n}x^{n} + a_{n-1}x^{n-1} + ... +a_{1}x+a_{0} \mid a_{i} \in F\}$ is ring of polynomials over F and $f(x)$ is irreducible polynomial in $F[X]$ ...
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45 views

Quicker way of finding minimal polynomial over Q than solving systems of equations?

I was trying to find the minimal polynomial of $\sqrt{i + \sqrt{2}}$ over $\mathbb{Q}$. I went as follows: $a = \sqrt{i + \sqrt{2}} $ $a^{2} = i + \sqrt{2} $ $a^{4} = 1 + 2\sqrt{2}i $ $(a^{4} - ...
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Is this a correct argument for an irred. polynomial of deg $m$ stays irred. over a field extension of deg $n$ if $n$ and $m$ are co-prime?

I have the following question: Let $F$ be a field and $K$ an extension field of $F$ of degree $n$. Let $f(x)\in F[x]$ be an irreducible polynomial of degree $m$. Suppose $n$ and $m$ are relatively ...
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78 views

Are all prime-order Fibonacci polynomials irreducible over $\Bbb{Z}$?

The Fibonacci polynomials are defined recursively as functions of one variable: $$ F_0(z) = 0 \\ F_1(z) = 1 \\ \forall n > 1: F_n(z) = zF_{n-1}(z) + F_{n-2}(z) $$ Thus for example $F_5(z) = 1+3z^2 +...
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48 views

Methods of determining irreducibility of a polynomial in a large finite field

Given a finite field $\mathbb{F}_p$ and some polynomial $f(x)\in\mathbb{F}_p [x]$. What are some of the methods of determining the irreducibility of $f(x)$? I feel like there are many theorems that we ...
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243 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
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448 views

Easy method of determining if a polynomial over $\Bbb{Z}$ has any quadratic factors with rational coefficients

There is an easy method of determining whether a monic polynomial $$\sum_0^n a_k x^k$$ with all $a_k \in \Bbb{Z}$ and $a_n = 1$ has any integer roots. At least it is easy if you can factor the ...
3
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0answers
98 views

Root finding using Galois theory

Is there a method in Galois theory that, say given an $n$th degree polynomial with integer coefficients $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ and $\alpha_1$ is a root of said polynomial, gives ...
3
votes
0answers
61 views

$x^5+a$ is reducible in $\mathbb{Z}_5[x]$ for each $a\in\mathbb{Z}_5$

Question: Show that $x^3+a$ is reducible in $\mathbb{Z}_3[x]$ for each $a\in\mathbb{Z}_3$, and that $x^5+a$ is reducible in $\mathbb{Z}_5[x]$ for each $a\in\mathbb{Z}_5$ So I got these two as my ...
3
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0answers
47 views

The irreducibility of the polynomial $x^p\pm px-t$, where $t$ is an integer, $p$ is a prime number.

Let $f(x)=x^p+px+1$, where $p$ is an odd prime. Prove that $f(x)$ is irreducible. This is an exercise of a course (linear algebra, the first chapter focus on the polynomial rings), and the exercise ...
3
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0answers
46 views

When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
3
votes
0answers
76 views

How to prove how many ireducible polynomials are in a polynomial ring over a finite field.

Can someone help me out with some problem concerning the cardinaliy of ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$? Here first why the problem appeared, I want to prove that $card(max(p\...
3
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98 views

Irreducibility of a family of univariate polynomials

While doing work on another problem, I came across the following family of polynomials: $$P_e(x) := ex^{2e} - x^{2e-1} - x^{2e-2} - \cdots - x + e\in \mathbb{Z}[x],$$ where $e\geq 1$ is an integer. ...