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Questions tagged [irreducible-polynomials]

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

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37 views

Show some polynomial satisfies Eisenstein's Criterion

Consider a polynomial $$f(X) = X^{(p-1)p^{n-1}} + X^{(p-2)p^{n-1}} + \cdots + X^{p^{n-1}} + 1$$ Now I need to show $f(X+1)$ safeties Eisenstein's criterion. My argument is that $$f(X+1) = (X+1)^{...
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1answer
31 views

Irreducibility of polynomials over some quadratic fields

Prove the following statements: a) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{2})$ b) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{-3})$ c) The polynomial $x^3-x-...
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2answers
24 views

Is $x^3-1$ reducible over $\mathbb{Q}$ [duplicate]

In other discussions, e.g. here it is claimed that a polynomial in a field $K$ with degree greater than $1$, having a root in $K$, must be reducible. So by this criterion $X^3-1$ would be reducible ...
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Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
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2answers
39 views

$f(x) = x$ in $(\mathbb{Z}/6\mathbb{Z})[x]$ factors as $(3x+4)(4x+3)$

This question has a lot of parts so I'll post each part separately. First, show $f(x) = x$ in $(\mathbb{Z}/6\mathbb{Z})[x]$ factors as $(3x+4)(4x+3)$ I am trying long division. I cannot divide $x$...
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2answers
27 views

Units, Primes and Irreducibles

How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$? Thank you.
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Degree of splitting field of $X^4+2X^2+2$ over $\mathbf{Q}$

Find the degree of splitting field of $f=X^4+2X^2+2$ over $\mathbf{Q}$. By Eisenstein, $f$ is irreducible. By setting $Y=X^2$, we can solve for the roots: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
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1answer
18 views

Degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$

What is the degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$ ? I note that $\cos(2\pi/8) + i \sin(2\pi/8)$ is a root of $x^8-1$. $x^8-1$ can be factored into $x^8-1 =(x^4+1)(x^2+1)(x+1)(...
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1answer
51 views

Check the properties of given polynomial of order $7$

Consider $$f(x)=x^{7}-105x+12$$ Then which of the following is\are correct? $f(x)$ is reducible over $\Bbb Q$ There exists an integer $m$ such that $f(m)= 105$ There exists an integer $m$...
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1answer
132 views

Is $x^4+4x-1$ irreducible in the field $Q[\sqrt{-7}]$?

I already know this is irreducible over $\mathbb{Q}$. Right now, my general plan is to show there is no root in $\mathbb{Q\sqrt{-7}$ (thus no linear factor). And then show that two quadratic factors ...
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1answer
22 views

Show that $x^p - m$ is irreducible for prime $p$ and $m \in \mathbb{Q^{\times}}\setminus \left(\mathbb{Q^{\times}}\right)^p$ [duplicate]

I'm stuck if $m$ is not a prime or has a single prime divider (Then using Eisenstein's criterion), e.g, $m=4$ and $p=5$. Any suggestions?
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1answer
25 views

Units, Prime Elements and Irreducible elements for polynomials

Are there any general steps or things to look out for when identifying the unit elements, irreducible elements and prime elements in ring of polynomials e.g. $\mathbb{Z}_4[x]$, $\mathbb{C}[x]$, $\...
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2answers
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If a monic rational polynomial of degree $p-1$ has $p$-th root of unity as a root, is it the cyclotomic polynomial?

If a monic rational polynomial of degree $p-1$ has a $p$-th root of unity as a root, where $p$ is prime, does that make it the cyclotomic polynomial $x^{p-1}+...+1$? I think this is the same as ...
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Factoring of polynomial over the rationals (quick question)

Okay this might be a stupid question, but couldn't find any clear explanation on the internet, so here goes. If I factor a polynomial into products of irreducible polynomials, are the products of ...
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1answer
97 views

Irreducible Polynomials In $F_3$

Let us say I have some irreducible polynomials in $F_3$ $$p(x) = x^3 + 2x + 2$$ and $$p(x) - 1 = x^3 + 2x + 1.$$ Now, using the power of Maple and Wolfram Alpha, we can check that $$p(x^{13}) = x^{...
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4answers
77 views

Using Eisenstein criterion show that $x^3+x^2-2x-1$ is irreducible? [duplicate]

I tried for few primes but they do not satisfy Eisenstein criterion.Also is there any approach other than brute force with the help of which we can find that prime.
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1answer
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Show that a polynomial is irreducible in $\mathbb { Z } [ i \sqrt { 5 } ]$ but not in $\mathbb { Q } [ i \sqrt { 5 } ]$

To prove : Show that the polynomial is irreducible in $3 + 2 t + 2 t ^ { 2 }$ over $\mathbb { Z } [ i \sqrt { 5 } ]$ but not over $\mathbb { Q } [ i \sqrt { 5 } ]$. Solution : The roots of the ...
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Showing $f(t) = t^4 + 2t^2 + 9$ is reducible over $\Bbb Q$

So, as a part of a homework problem, I'm tasked with showing that $f(t) = t^4 + 2t^2 + 9$, despite having no roots in the rationals, is reducible over them. The former task - showing a lack of ...
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1answer
46 views

Existence of irreducible polynomials with certain criteria

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements, where $q$ is an odd prime power. The question is as follows: Does there exists $a\in \mathbb{F}_{q}^*\setminus (\mathbb{F}_{q}^*)^2$, such ...
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2answers
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polynomial units in field

I am new on this topic and would like to have some clarifications. We were given a definition and theorem as follows: Definition : A polynomial $p$ in $F[x]$ is irreducible if $p$ is not a unit of $...
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2answers
200 views

Why is $(X^5-1)/(X-1)$ the minimal polynomial for $e^{2\pi i/5}$, the fifth root of unity, over $\Bbb Q$?

I have some trouble understanding an example from my reader$^1$. We have the fifth root of unity $\zeta_5=e^{2\pi i/5}$, so by definition $\zeta_5^5=1$. Then I want to find its minimal polynomial ...
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111 views

Prove that $f(x)$ and $g(x)$ do not have any roots in common.

Suppose that $a(x)f(x) +b(x)g(x) = 135$ where $a(x), b(x), f(x)$ and $g(x)$ are polynomials over $F$. Prove that $f(x)$ and $g(x)$ do not have any roots in common. Any help is appreciated; thanks!
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1answer
40 views

Maximal ideals of $\Bbb F_2[x]$

Prove or Disprove: $\Bbb F_2[x]$ has uncountably many maximal ideals For every integer $n$, every ideal of $\Bbb F_2[x]$ has only finitely many elements of degree $\leq n$. The first ...
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1answer
58 views

Proving that there are infinitely many irreducible polynomial in $\mathbb{Z}_{5} [x]$ [duplicate]

Pinter's textbook "A book of abstract algebra" asks to prove the following: There are infinitely many irreducible polynomials in $\mathbb{Z}_{5} [x]$. Here's my attempt to prove it any field $\...
3
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2answers
213 views

Irreducibility of a simple polynomial

For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $\mathbb{Q}$. What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 \mod 4$, ...
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1answer
50 views

Factorise $x^{111}+9x^{74}+27x^{37}+27$ in $\mathbb{Z}[x]$

I am having trouble with the following. Factorise $x^{111}+9x^{74}+27x^{37}+27$ in irreducible factors in $\mathbb{Z}[x]$. I did not find it to be an Eisentein polynomial and trying to find zeros ...
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1answer
34 views

Direct sum factorization of polynomials

I have been recently reading the paper "Mixed finite elements for second order elliptic problems in three variables" by Brezzi et. al. I noticed the claim in the proof of Lemma $2.1$, which basically ...
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1answer
71 views

Reducible and Irreducible polynomials over $\Bbb Q$

Consider polynomials of the form: $$x^r-(1-x)^k,$$ for $r,k\ge2.$ $x\in(0,1).$ When $r=k$ the polynomial seems to be reducible, except at $r=k=2.$ Do the irreducible and reducible polynomials ...
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1answer
40 views

Prove that the Galois group of $f\left(x\right)$ over $\mathbb{Q}$ is not a simple group.

Let $f\left(x\right)\in\mathbb{Q}\left[x\right]$ be an irreducible polynomial of degree $n>2$ which has $n-2$ real roots and exactly one pair of complex roots. Prove that the Galois group of $f\...
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2answers
82 views

Irreducibility of $x^5-10x^4+55x^3-110x^2+184x-60$ in $\mathbb{Q}[x]$

Let $g(x)$ be the above polynomial, I know that this polynomial is indeed irreducible, but I'm not entirely sure how to show it since the reduction mod $p$ test gives $g(k) \neq 0 \enspace \forall \...
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2answers
82 views

Prove $x^n+x^{n-1}+2$ is irreducible in $\mathbb{Q}[x]$ for $n\geq 2$

I want to prove $f(x)=x^n+x^{n-1}+2$ is irreducible in $\mathbb{Q}[x]$ for $n\geq 2$. I think I'm very close to doing this, and I have used the following theorem: Generalized Eisenstein criterion: ...
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1answer
83 views

Why is this polynomial irreducible at mostly prime exponents

I'm trying to determine when this polynomial is reducible and irreducible over $\Bbb Q$ : $$x^n+(1-x)^n.$$ I checked up to $n=100.$ This polynomial is irreducible for: $$n=2,3,4,5,7,8,11,13,16,17,...
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Why is $p_{3}(x) = 9x^{2}-3=3(x\sqrt{3}-1)(x\sqrt{3}+1)$ reducible over integers?

I am learning Irreducibility of polynomials, and I am reading this entry of Wiki. However, I am confused why the polynomial $p_{3}(x) = 9x^{2}-3=3(x\sqrt{3}-1)(x\sqrt{3}+1)$ is reducible over integers....
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2answers
33 views

What are the prime ideals of the polynomial ring $\mathbb{R}[x]$?

The exact question is: And the solution is What I don't get in this solution is the circled part. Why are there no other irreducible polynomials in $\mathbb{R}[x]$? Isn't $x^4+1$ irreducible in $\...
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3answers
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How can I factor $x^{12}+x^{11}+\cdots+x+1$ in $\mathbb F_3[x]$?

I can prove that $\mathbb F_3[x]$ is a UFD, so $f(x)=x^{12}+x^{11}+\cdots+x+1$ can be factored. And because neither 0, 1, nor 2 is a root of $f(x)$, all factors are more than 1 degrees. But I don’t ...
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0answers
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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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3answers
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Is this polynomial irreducible? $h(x, y) = x^2\ − y^3 \in \mathbb{Q}[x,y]$

$h(x, y) = x^2\ − y^3 \in \mathbb{Q}[x,y]$ I am trying to figure out if this polynomial is irreducible. I believe it is, but I am not sure how to show this. Any help would be great!
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1answer
32 views

Is $x$ always irreducible in the quotient ring

Let $I$ be a prime ideal in $\mathbb{C}[x,y]$. Then is it true that $x + I$ is always either a unit, or irreducible? I am trying to show that $x + I$ is irreducible for $I =(y^2-x^3-1)$, however I ...
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0answers
66 views

Show that an element is irreducible but not prime.

Show that in the ring $\mathbb Q[x, y]/(x^3 -y^2)$ the element $x +(x^3 -y^2)$ is irreducible but not prime. Not sure how to show this. I know that $(x^3 -y^2)$ is a prime ideal but I cannot ...
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a question for the irreducibility test for degree 2 or 3 polynomial

I have a question for the irreducibility test for degree 2 or 3 polynomial. The test states: "Let F be a field, f(x) ∈ F[x], and degf(x) = 2 or 3. Then f(x) is reducible over F if and only if f(x) has ...
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0answers
38 views

Does the polynomial $z^{n+1} + w^{n+1}$ factor as $(z^n + w^n)f(z,w)$?

Does the complex polynomial $z^{n+1} + w^{n+1}$ factor as $(z^n + w^n)f(z,w)$? I don’t have any conditions on $f$ apart from that it must be continuous. If the answer is yes, I would appreciate ...
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1answer
30 views

Questions about proof of existence of roots of $f$ in $K[X]/(f)$

Let $f \in K[X]$ with $deg(f)\geq 1$. Then there exists an algebraic field extension $L/K$, such that $f$ has a root in $L$. Proof: WLOG we can assume that $f \in K[X]$ is irreducible. Since $...
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1answer
27 views

Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$ K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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2answers
54 views

Characteristic polynomial of $v\mapsto X\cdot v$ over $V := K[X]/(f_1)\bigoplus\dots\bigoplus K[X]/(f_n)$

I'm doing a study on module theory and I'm working on the following problem: Let $K$ be a field and let $f_1, \dots, f_n \in K[X]$ be monic polinomials. Consider the $K[X]$-module $$ V := K[X]/(f_1)\...
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1answer
64 views

Irreducible Polynomial of Galois field

We know that one irreducible polynomial on $\Bbb{Z}_2[x]$ is $x^8 + x^4 + x^3 + x + 1$. How to check that it is irreducible? And how to generate irreducible polynomial for any degree?
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2answers
64 views

Factor $X^4 + 3$ into irreducible factors in $F_7[X]$

I am not quite sure where to start with this problem. I am new to polynomial rings and want to learn how to factor polynomials in polynomial rings made of fields. Factor $X^4 + 3$ into irreducible ...
3
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0answers
62 views

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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1answer
60 views

Understanding Lang's Chapter 6 Theorem 9.1

For the past week, I've been trying to dissect and make rigorous my understanding of Lang's theorem, which says that: Theorem 9.1 Let $k$ be a field and n an integer $\geq$ 2. Let $a \in k, a \neq ...
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1answer
67 views

Splitting field of separable polynomial is Galois extension

Definitions: $f$ is separable if every irreducible factor has distinct roots. $E/F$ is a Galois extension if the fixed field of the Galois group Gal$(E/F)$ is $F$ I would like to prove the following ...
5
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0answers
88 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) \...