Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [irreducible-polynomials]

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

0
votes
1answer
19 views

Irreducibility of $x^n-a^n$ on $F(a^n)$ where $a$ is transcendental on $F$

I'm looking for a short proof of the following : If $F$ is a field where $a$ is transcendental, $x^n-a^n$ is irreducible on $F(a^n)$. I've managed to prove it in a tedious way but I feel like there ...
2
votes
1answer
361 views

$f$ irreducible over $\mathbb{Z}_{p}$ implies $f$ is irreducible over $\mathbb{Q}$

Let $f \in \mathbb{Z}[x]$ be a non-constant polynomial and let $p$ be a prime number which is not a divisor of the leading coefficient of $f$. I need to prove that if $f$ is irreducible over $\mathbb{...
2
votes
4answers
703 views

Simple extension of $\mathbb{Q} (\sqrt[4]{2},i)$

I know the simple extension of $\mathbb{Q} (\sqrt[4]{2},i)$ is $\mathbb{Q} (\sqrt[4]{2}+i)$, but how do I show this? One direction is easy, but I have trouble with the other direction.
0
votes
0answers
24 views

The polynomial $aX^2+bX+c$ of degree $2$ in a field $K$ with char$(K)\neq 2$ is irreducible if and only if $b^2-4ac$ is not a square in $K$.

I have to prove the following: The polynomial $aX^2+bX+c$ of degree $2$ in a field $K$ with char$(K)\neq 2$ is irreducible if and only if $b^2-4ac$ is not a square in $K$. I have never worked with ...
0
votes
1answer
27 views

If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Then $p(x) | f(x)$

If $p(x)$ is the minimal polynomial of $\alpha$, and $f(x) \in F[x]$, $f(\alpha) = 0$, Show: $$p(x) | f(x)$$ I'm not sure if it's a direct consequence of the proof that minimal polynomial of an ...
0
votes
0answers
48 views

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$ I think $p$ is supposed to be a prime for the only ...
0
votes
0answers
11 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...
4
votes
1answer
157 views

Explain proof of irreducibility of $x^{p-1} + 2x^{p-2}+ \dots +(p-1)x + p$

This is a question from Putnam and Beyond, and I have a question about the proof. The question is: Show $x^{p-1} + 2x^{p-2} + 3x^{p-3} + \dots + (p-1)x + p$ is irreducible over $\mathbb{Z}[X]$. ...
6
votes
1answer
124 views

Family of irreducible polynomials

Consider the following family of polynomials $$P_n(X)=\sum_{i=0}^n(n+1-i)X^i,\,n\ge 1$$ Let’s write down the first few $$ \begin{align} P_1(X)=&X+2\\ P_2(X)=&X^2+2X+3\\ P_3(X)=&X^3+2X^2+...
4
votes
1answer
42 views

Degree 2 Recurring monic polynomials

Consider a monic polynomial $x^2+ax+b=0$, with real coefficients. If it has real roots $p$ and $q$, such that $p\leq q$, then you construct a new monic polynomial as $x^2+px+q=0$. If this polynomial ...
0
votes
1answer
23 views

Find the irreducible polynomial over $Q$(linear combination of primitive cubic roots)

Hi I'm student who just started the algebra. There are some question that bothering me. Let $\omega = e^\frac{2\pi i }{7}$ I've already known the $irr(\omega,Q) = w^6 +w^5 +w^4+w^3+w^2+w+1$ (...
0
votes
3answers
35 views

Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$ [duplicate]

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the ...
1
vote
0answers
63 views

Polynomial that has no rational roots yet has roots modulo every integer

I want to characterize when the polynomial $p(x) = (x^{2} - a) (x^{2} - b) (x^{2} -ab)$ has a root modulo every integer, yet doesn't have an integer root. I worked out the condition when '$a$' and '$...
1
vote
3answers
89 views

Proving $f(x)=x^4+4x^3+3x^2+7x-4$ is irreducible

Here's my attempt, I'm almost there but I'm stuck: Using a hint, I wrote the modular reduction: Reducing the coefficients modulo $2$ gives: $\left [ f \right ]_2=x^4+x^2+x=x(x^3+x+1)$. Reducing ...
0
votes
1answer
45 views

How to show a polynomial is reducible/irreducible in a ring

I have to show that these polynomials is reducible or irreducible in the given ring. $a)$ $2x^3 − 5x^2 + 6x − 2$ in $\mathbb{Z}[x]$ $b)$ $x^4 + 4x^3 + 6x^2 + 2x + 1$ in $\mathbb{Z}[x]$ I think I ...
0
votes
0answers
17 views

Proof that a polynomial is irreducible over Q [duplicate]

How could I prove that $f=[(X-1)(X-2)...(X-n)]^{2}+1$ is irreducible over $\mathbb{Q}[X]$ ? I've tried to use Pólya's query of divisibility: If $f$ in $\mathbb{Z} [X]$ is a polynomial of degree $...
2
votes
1answer
51 views

when is the $n$-th cyclotomic polynomial irreducible over $\mathbb{R}$

It is well known that the cyclotomic polynomials $\Phi_n(x)$ are irreducible over the field of rationals $\mathbb{Q}$. I am curious about their reducibility over the real numbers $\mathbb R$. We have ...
3
votes
3answers
89 views

Prove that $x^2 − 2$ is irreducible over $\mathbb Q (\sqrt 3)$

Prove that $x^2 − 2$ is irreducible over $\Bbb Q(\sqrt 3)$. I was originally trying to use the fact that if $K=\Bbb Q(\sqrt 3)[x]/(x^2-2)$ and $[K:\Bbb Q(\sqrt 3)]=2$ then $x^2-2$ is irreducible over ...
3
votes
2answers
106 views

Irreducible polynomial. [closed]

Prove that polynomials $p(x)=\left((\prod_{i=1}^{n} (x-a_i))^{2}\right)+1$ and $q(x)=\left(\prod_{i=1}^{n} (x-a_i)\right)-1$ are irreducible over $z$ ,where $a_i $ ' s are distinct integers .
0
votes
4answers
86 views

Irreducible Polynomial Field Extensions with Root $\cos \frac{2\pi}{7}$

Show that $\theta = \frac{2k\pi}{7}$ satisfies the equation $\cos 4\theta − \cos 3\theta =0$ for each integer $k$. Hence find an irreducible polynomial over $\Bbb Q$ with $\cos \frac{2\pi}{7}$ as a ...
1
vote
1answer
62 views

How to prove irreducibility of this polynomial?

Let $p,q$ be primes. Prove that $y^{n }-p$ is irreducible over $\mathbb{Q}(\sqrt[n ]q) $ . I have tried for some time, but still feel confused about how to prove it. Can anyone help me?
0
votes
1answer
32 views

How do I show that $y^2 - x(x-1)(x+1)$ is irreducible in $\mathbb{R}[x, y]$?

I tried writing this polynomial as a product of two polynomials $g, h$ of degree 2 and 1, respectively, and tried to arrive at a contradiction by multiplication of their coefficients. However, I ...
0
votes
3answers
85 views

Show that the polynomial $7X^5 + 71X^3 - 9$ is irreducible in $\mathbb{Z}[X]$

Show that the polynomial $f(X) = 7X^5 + 71X^3 - 9$ is irreducible in $\mathbb{Z}[X]$ My solution: Using the irreducibility test: "Reduction Mod p Test" $f(X)$ is clearly primitive and that the ...
0
votes
1answer
31 views

Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible

Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$...
3
votes
3answers
110 views

The real positive root of $9x^5+7x^2-9=0$

$$9x^5+7x^2-9=0$$ How do we evaluate the roots of the given polynomial? We're asked to find its real positive zero. What I tried doing: Let $$f(x)=9x^5+7x^2-9$$ Using Descartes' rule of signs, I ...
0
votes
0answers
30 views

Irreducible polynomial with root implies least degree

Let $\alpha$ be an element of an extension field $K$ of the field $F$ that is algebraic over $F$. Then, if $f$ is an irreducible element of $F[x]$, and $\alpha$ is a root of $f$, then $f$ is the monic ...
0
votes
3answers
29 views

Irreducibility of $2x^3-3x^2+6$ in $\mathbb Q[x]$

The following statement is given, i need to check whether its true\false. There exists a subfield $F$ of $\mathbb{C}$ such that $F\not\subseteq\mathbb R$ and $$F \cong \mathbb Q[X]/(2X^3 − 3X^2 + 6)....
14
votes
5answers
15k views

Irreducible polynomial means no roots?

If a polynomial is irreducible in $R[x]$, where $R$ is a ring, it means that it does not have a root in $R$, right? For example, to say that a polynomial $f(x)\in\mathbb Z[x]$ is irreducible in $\...
1
vote
2answers
882 views

Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are those?

I don't understand the three claims that some $F_a[X]/(p(x))$ has some $n$ elements in the following text (from Adkins' Algebra): For example Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are ...
0
votes
1answer
59 views

Show $x^{p}+a$ is reducible in $\mathbb{Z}_p[x]$

first time taking abstract algebra stuck on this question and have no idea how to go about it. Show that $x^{p}+a$ is reducible in $\mathbb{Z_{p}}[x]$: for each $a$ in $\mathbb{Z_{p}}$, where $p$ is a ...
-2
votes
1answer
32 views

How could I prove that $x^2+1$ and $x^4+1$ are irreductible over $\mathbb{Z}/2 \mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$? [duplicate]

How could I prove that $x^2+1$ and $x^4+1$ are irreductible over $\mathbb{Z}/2 \mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$? Theorem : Let $A$ an integral domain and $I$ a proper ideal of $A$. If $f(x) \...
4
votes
2answers
496 views

How can I prove that $x^4-2x^2+9$ is irreducible over $\mathbb{Q}$? [closed]

How can I prove that $x^4-2x^2+9$ is irreducible over $\mathbb{Q}$? Is there any general way to prove irreducibility of polynomials?
5
votes
4answers
403 views

How can I show that the polynomial $p = x^5 - x^3 - 2x^2 - 2x - 1$ is irreducible over $\Bbb Q$?

I've tried a few "criteria" to check if this is irreducible. According to Maple it only has one entirely real root which I suspect is not rational but I can't prove it so I'm attempting to check if $p$...
5
votes
3answers
134 views

Polynomial factorization into irreducibles over $\mathbb{Q}[x]$

I need to find irreducible factors of $f(x)=x^4+3x^3+2x^2+1$ in $\mathbb{Q}[x]$ and explicitely prove that these factors are indeed irreducible. I believe we can't reduce $f(x)$ any further but I ...
1
vote
3answers
95 views

Find the minimal polynomial of $i\sqrt{-1+2\sqrt3} \in \Bbb C$ over $\Bbb R$

I need to find the minimal polynomial of $i\sqrt{-1+2\sqrt3} \in \Bbb C$ over $\Bbb R$ and prove the polynomial that I find is actually the minimal polynomial. I know that the $\Bbb R$-basis for $\Bbb ...
2
votes
1answer
55 views

Why we ignore the other elements of $F_{p^k}$ when we check if any element of it is a root of multiplicity greater than $1$ in $p(x)$ or not?

$p(x) = x^4 + x +6$ . I was to find if there is any root of multiplicity greater than $1$ in the field of characteristics $p$. I was suggested to check every element of the $F_p$. But a filed ...
3
votes
1answer
93 views

How to prove this polynomial is irreducible over $\mathbb{Q}$

$f(x)=x^{n}+x+p$ where $p\geq 3$ is a prime.Prove that $f(x)$ is irreducible over $\mathbb{Q}$. My try:Let $y=x+m$ for some $m$ and try to use Eisenstein's Criterion.But I can't find the suitable $m$....
0
votes
1answer
22 views

$y^2z - x^3$ is irreducible and with one singularity

Give an example for an irreducible cubic curve in $\mathbb{C}\mathbb{P}^2$ with exactly one singular point. It is easy to check that $y^2z - x^3$ has only [0,0,1] as a singularity. But how to show it ...
8
votes
3answers
5k views

Irreducibility of $X^{p-1} + \cdots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $X+1$, ...
0
votes
2answers
51 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)^{...
1
vote
1answer
36 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-...
2
votes
1answer
49 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 ...
0
votes
2answers
29 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 ...
0
votes
0answers
18 views

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 ...
3
votes
4answers
818 views

Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$.

I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare. Thanks in advance
-1
votes
1answer
74 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?
0
votes
2answers
42 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 $x$...
6
votes
3answers
149 views

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: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
2
votes
2answers
29 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.
1
vote
1answer
21 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)(...