Questions tagged [irreducible-polynomials]

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

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64 views

Proving The Irreducibility Of $X^{2p}+pX^n-1$ Over $\mathbb{Z}[X]$

The following is an exercise in Victor V. Prasolov's Polynomials (Second edition, Page no. $74$, exercise $2.10$) Problem: Let $p>3$ be a prime and $n<2p$ a natural number. Prove that $$P(X)=X^{...
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0answers
23 views

Constructing an irreducible polynomial $f(x)$ over $\mathbb{Q}$ of degree $n$ s.t. $f$ has $n$ real roots

By Eisenstein's irreducibility criterion we can construct an irreducible polynomial over $\mathbb{Q}$ of any positive degree (For example, $x^n+2$), but how to make it that it has n roots over $\...
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32 views

Let $R=K[x,y,z]/(x^2-yz)$, where $K$ is a field. Show that $R$ is an integral domain, but not a unique factorization domain.

Let $R=K[x,y,z]/(x^2-yz)$, where $K$ is a field. Show that $R$ is an integral domain, but not a unique factorization domain. My idea: Since $K$ is a field, and I want to show $R$ is an integral domain....
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3answers
48 views

Relation between irreducible polynomial and dimension

I found the following in my textbook: Consider the field extension $\mathbb{Q} \subset \mathbb{Q}(\alpha_1)=:K$ ($\alpha_1$ is a root). Since $f=x^4-2$ is irreducible by Eisenstein's criterion or by ...
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23 views

Problem with Irreductible Polynomials in $k[x]$, $k$ be an field.

I don't have idea, how i can prove this: Let $k$ be a field, and let $f(x) = a_{0} + a_{1}x + \cdots + a_{n}x^{n} \in k[x]$ have degree $n$ and nonzero constant term $a_0$. Prove that if $f(x)$ is ...
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1answer
59 views

Prove the irreducibility of the polynomial

Let $p$ be a prime and set $g = X^2 + X - 1\in \mathbb{F}_p[X]$. Prove the following: $$\text{$g$ is irreducible in $\mathbb{F}_p[X]$ iff $g$ is irreducible in $\mathbb{F}_{p^3}[X]$}$$ One side is ...
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25 views

Points in “general position” with respect to polynomials

I have a question about points in general position. In my mind I am imagining that I am working over the field $\mathbb{C}$ and in dimension $d=3$, but I am stating the problem with slightly more ...
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pairs of polynomials and \gcd

Let $Q=x^h$, $h>0$ is an integer, and $P$ is an irreducible polynomial over the field $\mathbb{Q}$ of rational numbers of degree $>h$ such that $P(0)\neq -1.$ Can we prove that for all $n>1$ $...
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27 views

How to find the generators of the multiplicative group of a finite field

Input: p (prime number), n (positive number) Output: g ( generator ) I have just found an irreducible polynomial over $F_p[x]$. Now I must find all generators of the multiplicative group from this ...
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Proof? of irreducibility of cyclotomic polynomial

Let $\omega = e^{\frac{i2\pi}{n}}$. I am trying to show that the minimal polynomial of $\omega$ over $\mathbb{Q}$ is the cyclotomic polynomial, that is the polynomial whose roots are the primitve $\...
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3answers
91 views

Showing $x^4 + 2$ is irreducible in $F_5[x]$

There is a similar question like this here but I don't understand the solution. Since this is a fourth degree and since it has no root in $F_5$, it can only have a quadratic reduction. But how do we ...
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1answer
60 views

Basic polynomial questions

Find a rational polynomial such that $$P(n)=1\cdot 2+ 2\cdot 3+\cdots + n\cdot(n+1).$$ for all positive integers $n$ (edited). Does there exist an integer polynomial of this form? Ive found that $P(X)...
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1answer
66 views

Is the ideal $x^3-y^5 \subseteq \mathbb{C}[x,y]$ prime? Is it maximal? [closed]

I have some idea like I is a maximal ideal of a commutative ring R iff R/I is a field. but not able to formulate for this case. first, I thought about the irreducibility of ideal $x^3-y^5 \subseteq \...
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1answer
41 views

Why does the polynomial splitting implies existence primitive root of unity in $\Bbb{F}_{p^2}$?

This question refers to WimC's answer to this question. Consider the cubic congruence problem: $$ f(x) := x^3 - x^2 - 2x + 1 \equiv 0 \pmod{p} $$ We want to know for which $p$ does $f(x)$ splits. The ...
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27 views

How to calculate the irreducible polynomial in galois field

I have a an expression (x^3 + x^2 + 1) / (x^6 + x^5) in GF(2^8) and its primitive polynomials (0,1,3,4,8) How to deal with this ...
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Find prime $p$ which $x^3 + ax + b \equiv 0 \pmod{p}$ has a root with square discriminant

This is a follow-up to this question. As mentioned in the comment and the answer, it's difficult to classify all $p$ which $ax^3 + bx^2 + cx + d \equiv 0 \pmod{p}$ has a root for arbitrary $a,b,c,d \...
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2answers
84 views

Let $K = \mathbb{F}_3[T]/(T^3-T+1)$, what would be an irreducible polynomial in $K[X]$ of degree $13$?

Let $K = \mathbb{F}_3[T]/(T^3-T+1)$. I'm trying to find an explicit polynomial $f \in K[X]$ that is irreducible of degree $13$. My first attempt was to note that since $T^3-T+1$ is irreducible over $\...
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1answer
43 views

Number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$

Let $p,n$ be two odd prime numbers. I want to show that the number of irreducible factors of $x^{p^n + 1} - 1$ over $\Bbb F_p$ is $$N = \frac{p^n-p}{2n} + \frac{p-1}{2} + 2$$ I know that this is equal ...
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1answer
33 views

What is the method for finding prime, irreducible, invertible elements of the Z/nZ [closed]

For example, how can I find prime, irreducible and invertible elements of the Z[6], and which element is associated with $\overline3,\ $
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1answer
67 views

Existence of solution to $ax^3 + bx^2 + cx + d \equiv 0 \pmod{p}$

Given a polynomial $f(x) = ax^3 + bx^2 + cx + d \equiv 0 \pmod{p}$, $a \not\equiv 0 \pmod{p}$, I would like to classify all primes $p$ so that there exists $\alpha \in \Bbb{F}_p$ in which $f(\alpha) \...
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1answer
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If a polynomial is irreducible and nonconstant over a finite field, it has a multiple root iff it is in the variable $x^p$

I am a very basic field theory question. I must be mixing up a theorem here, but I am unsure which. My goal here is to determine if there exists an inseparable, irreducible polynomial in a finite ...
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2answers
33 views

Finding an irreducible polynomial over the rationals

I'm very confused by a homework question. "Find the irreducible polynomial for $ \sin{2\pi/5}$ over Q. I found that $16t^{4}-20t^{2}+5=0$ but this is not monic? This is also irreducible by ...
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2answers
48 views

$x^2+3x+1$ as irreducible polynomials [closed]

can you help! Write $x^2+3x+1$ as a product of irreducible polynomials in $\Bbb Z_5[x]$.
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All roots of a polynomial in ring $F_2+uF_2+u^2F_2$, where $u^3=0$

Let $R=F_2+uF_2+u^2F_2$, where $u^3=0$, be a finite commutative ring. So $R=\{0,1,u,v,uv,u^2,v^2,v^3\}$, where $v=1+u$, $v^2=1+u^2$, $v^3=1+u+u^2$, $uv=u+u^2$. It is well known that $$x^7-1=(x+v^3)(x^...
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0answers
75 views

Galois group of $x^4+tx+t\in F(t)[x]$

Let $F$ be a field of characteristic $\operatorname{char}(F)\ne2$. Prove that the polynomial $f(x)=x^4+tx+t\in F(t)[x]$ is irreduciable, and find its Galois group. Proving that the polynomial is ...
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Is there any website like OEIS for special polynomials

I would like to know if there is any kind of website (like OEIS) in which we can search for special known polynomials. For example, we put the coefficients of Legendre's and then the website gives us ...
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1answer
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$\gcd$ of polynomials over a finite field

Let $p$ be an odd prime number and $\mathbb{K}:=\mathbb{F}_{p^{t}}$ be a field of characteristic $p$. Let $u\in \mathbb{K}$ such that $T^2 -u$ is irreducible over $\mathbb{K}$. Prove that for all $n\...
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0answers
57 views

How to prove that $f(x)=x^7+7x^2+2$ is irreducible over $\mathbb{Q}$?

How to prove that $f(x)=x^7+7x^2+2$is irreducible over $\mathbb{Q}$? I have tried to use Eisenstein's Criterion. I pick $-2$,since $$(-2)^7+7(-2)^2+2\equiv 0~~ (\mod 7)$$ Let $y=x-2$,then $$f(x)=...
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2answers
36 views

Irreducibility of a Polynomial in Q[x]

let f(x)=2x^2-8 is a polynomial in Q[x].Although we can chech for reducibility by checking the zero of the polynomial in Q(because it is a field) and we see that x=2 is a zero of this. but if we write ...
2
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4answers
110 views

Is $X^8+3X^4-53$ irreducible over $\mathbb{Z}[X]$?

I want to determine whether or not the polynomial $X^8+3X^4-53$ is irreducible over $\mathbb{Z}[X]$. I noticed that it doesn't have integer (or rational) roots but I have no further ideas.
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21 views

Ring isomorphism preserving irreducibility

My Question is; (1) For a commutative ring $R$ with the unity and $a$ in $R$, $f: R[x]\to R[x]$ by $f(p(x))=p(x-a)$ for $p(x)$ in $R[x]$ is a ring isomorphism, (2) A ring isomorphism (between integral ...
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1answer
37 views

True or False questions regarding $𝔽_9$ with the irreducible polynomial $x^2 +2x+2$

Let $𝔽_9$ be constructed with the irreducible polynomial $x^2 +2x+2$. For $a,b \in 𝔽_3$ we write $ax+b \in 𝔽_9$ for $ab$. In our exam we had to find out whether the following are true or false. I ...
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1answer
40 views

Why is $x^4+x^2+1$ over $𝔽_2$ a reducible polynomial? What do I misunderstand?

I don't quite understand when a polynomial is irreducible and when it's not. Take $x^2 +1$ over $𝔽_3$. As far as I know, I have to do the following: 0 1 2 using $x \in 𝔽_3$ 1 2 2 using $p(x)$ I ...
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4answers
102 views

How do I prove that $x^2 + y^2 - 1$ irreducible is $\mathbb{R}[x,y]$

I have the polynomial $x^2 + y^2 - 1$ in the ring $R = \mathbb{R}[x,y]$. I have to prove that it's irreducible in $\mathbb{R}[x,y]$ and I want to do it without using Eisenstein's Criterion otherwise ...
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0answers
18 views

Algorithm to determine splitting field of polynomial?

I'm thinking about splitting fields and how to determine them in general. I couldn't find any general algorithm to determine the splitting field $L$ of an arbitrary irreducible polynomial $f$ over a ...
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0answers
43 views

Find decomposition of $x^8 -1$ over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_4$

Let us consider $x^8 -1$. I want to decompose it over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_4$. In $\mathbb{F}_3$ I have no problem, since I can use the cyclotomic cosets. In $\mathbb{F}_2$...
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1answer
61 views

When the polynomial $16(n+1)^2(p(x)-1)^3p(x)+1$ is a perfect square

Let $n$ be a positive integer. Let $p(x)$ be a positive polynomial with positive integer coefficients. I am asking when the polynomial $16(n+1)^2(p(x)-1)^3p(x)+1$ is a perfect square, i.e., there ...
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1answer
40 views

Is the following a correct proof that $t^4-32$ is irreducible over $\mathbb{Z}$?

I just want to check if the following proof is correct. Let $f(t)=t^4-32\in \mathbb{Z}[t]$. Then $$f(t)=(t-\sqrt[4]{32})(t+\sqrt[4]{32})(t-i\sqrt[4]{32})(t+i\sqrt[4]{32})$$ but $\sqrt[4]{32}=2\sqrt[...
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1answer
12 views

Question about generator of $K[x]$ using simple extension.

Let $K \subset \mathbb{C} $ be a field, and $K[x]$ the polynomial ring, $\alpha$ algebraic over $K$ and $L=K(\alpha)$. I am reading this book and it states that, since any element of $L$ is of the ...
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0answers
41 views

Irreducible polynomial in field extension (alternative proof)

I am interested in the following theorem. Let $L/K$ be a field extension of degree $n$. Let $P(X)$ be a degree $d$ polynomial that is irreducible over $K$. If $n$ and $d$ are relatively prime, then ...
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0answers
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Find splitting field and irreducible decomposition [duplicate]

I'm trying to solve the following exercise. I did the first two points( hope they're right), but have no idea on how to solve the last one. Let $f(x)= x^8 -x$ in $\mathbb{F}_3[x]$. Find: ...
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1answer
37 views

Specific detail in the irreducibility of $(x-1)(x-2)…(x-n)-1$ in the integers.

I am studying the seemingly official accepted answer to this question here. Here is a screen shot of the answer. If we assume that $p(x)$ is reducible in $\mathbb{Z}$, then we can assume a ...
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0answers
50 views

A problem regarding polynomials with only prime powers [duplicate]

$\mathbf {The \ Problem \ is}:$ Let, $f \in \mathbb Q[x]$ be a polynomial of degree $n \gt 0$ and let $p_1,p_2, \cdots p_{n+1}$ be distinct prime numbers . Show that there exists a non-zero polynomial ...
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0answers
42 views

List all monic irreducible polynomials of prime degree $p$ over $\mathbb{F}_p$

There are $p^{p - 1} - 1$ monic irreducible polynomials of prime degree $p$ over $\mathbb{F}_p$ by this post. The chance of picking one of them randomly is $\cfrac{p^{p - 1} - 1}{p^p} = \cfrac{1}{p} -...
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0answers
16 views

Irreducibility of Polynomials over Residue Class Ring

Given some polynomial $P \in \mathbb{Z}/m\mathbb{Z}[T]$, is there an easy way to check if $P$ is irreducible? My understanding is that I cannot use Eisenstein's criterion because, in general, $\mathbb{...
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2answers
67 views

How can I prove that a polynomial is irreducible over $\mathbb Z[x]$? [closed]

I am asked to prove that $$P(x) = x^6 + x + 1$$ is irreducible over $\mathbb Z[x]$. I tried using Eisenstein criteria by a doing a change of variable such as $x = y + a$ but I was unsuccessful.
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2answers
40 views

Differences between polynomial quotient rings $\mathbb{Z}_m[x]/(x^n+1)$ and $\mathbb{Z}_m[x]/(x^n-1)$

As based on the definition of the polynomial quotient ring $\mathbb{Z}_m[x]/(x^n+1) = \left\{a_{n-1}x^{n-1}+\cdots+a_1x+a_0:a_i\in\mathbb{Z}_m\right\}$, does that imply that $\mathbb{Z}_m[x]/(x^n+1)...
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1answer
25 views

All irreducible polynomials of degree 4 over a field of 4 elements [closed]

Find out all irreducible polynomials of degree 4 over a field of 4 elements. Help me, please
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1answer
30 views

Let $F=\Bbb{F}_3[x]/(f(x))$ and $\alpha=x+f(x) \Bbb{F}_3 \in F$, so that $f(\alpha)=0$ Prove that f(x) is irreducible

Let $f(x)=x^3+2x^2+1 \in \Bbb{F}_3[x]$. Let $F=\Bbb{F}_3[x]/(f(x))$ and $\alpha=x+f(x) \Bbb{F}_3 \in F$, so that $f(\alpha)=0$ Prove that f(x) is irreducible. Gauss' lemma: suppose $f \in \Bbb{Z}[x]$ ...
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0answers
36 views

Prove polynomial is irreducible in $Z_p[x]$.

Give $f(x)=a_0+a_1x+\ldots+a_nx^n$ and prime $p$ that $p \nmid a_n$ and $GCD(a_1,a_2,\ldots,a_n)=1$. Which one in two clause below is correct? (1):"If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then $f(...

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