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Questions tagged [irreducible-polynomials]

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

69
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7answers
15k views

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
29
votes
3answers
8k views

Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
49
votes
2answers
17k views

Number of monic irreducible polynomials of prime degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
32
votes
3answers
31k views

Find all irreducible monic polynomials in $\mathbb{Z}/(2)[x]$ with degree equal or less than 5

Find all irreducible monic polynomials in $\mathbb{Z}/(2)[x]$ with degree equal or less than $5$. This is what I tried: It's evident that $x,x+1$ are irreducible. Then, use these to find all ...
11
votes
3answers
11k views

How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such $...
44
votes
1answer
5k views

$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?

Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]: Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in F$. Then $...
16
votes
2answers
2k views

When is $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
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 $X+1$, ...
7
votes
2answers
845 views

Eisenstein Criterion with a twist

As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a ...
29
votes
2answers
3k views

Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$

I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-...
6
votes
4answers
12k views

How to find all irreducible polynomials in Z2 with degree 5? [duplicate]

I am totally lost on how to do this one. I am supposed to accomplish the following: Find all irreducible polynomials in $\mathbb{Z}_2[x]$ with degree $5$. I may use the fact that x, $x+1$ and $x^2+x+...
29
votes
3answers
2k views

Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a unique factorization domain $R$, say $$f(x)=a_0 + a_1x + a_2x^2 + \...
13
votes
1answer
6k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge |...
15
votes
5answers
1k views

$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$

I am trying to show that $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ where $n,m$ are positive integers. I showed that if $\gcd(n,m)$ is not $1$, then $X^n-Y^m$ is reducible. How to ...
11
votes
7answers
4k views

Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$

How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don't exactly know how that applies. Thanks for any input.
10
votes
1answer
577 views

Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

If $\mathbb{F}_q$ is a finite field with $q=p^m$ elements where $p$ is prime, if $a\in\mathbb{F}_p^\times$, ¿Which are the irreducible factors of $f_a=x^q-x-a$ ? Attempt: If $\alpha$ is a root for $...
14
votes
3answers
4k views

Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$. This answer to a related question ...
11
votes
3answers
652 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over $\...
4
votes
2answers
1k views

Galois group of irreducible cubic equation

The polynomial $f(x) = x^3 -3x + 1$ is irreducible, and I'm trying to find the splitting field of the polynomial. We've been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a ...
7
votes
4answers
470 views

Irreducibility issue [duplicate]

This is a homework question. Given $f(x)=x^{p-1}+x^{p-2}+\cdots+x+1$, where $p$ is any prime. Prove that $f(x)$ is irreducible over $\mathbb{Z}[x]$? Any idea, hint, etc? Hint given by my book was ...
20
votes
1answer
1k views

Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree [duplicate]

Question is to prove : Irreducibility of $(x-1)(x-2)\cdots (x-n)- 1$ over $\mathbb{Z}$ for all $n\geq 1$ Irreducibility of $(x-1)(x-2)\cdots (x-n)+ 1$ over $\mathbb{Z}$ for all $n\geq 1$ and $n\neq ...
26
votes
1answer
829 views

$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$

Suppose that $a_1,a_2, \cdots, a_n$ are $n$ different integers. Then $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$. I've no idea why it is true. Thanks very much.
14
votes
6answers
3k views

$x^2 +y^2 + z^2$ is irreducible in $\mathbb C [x,y,z]$

Is $x^2 +y^2 + z^2$ irreducible in $\mathbb C [x,y,z]$? As $(x^2+y^2+z^2)= (x+y+z)^2- 2(xy+yz+zx)$, $$(x^2+y^2+z^2)=\left(x+y+z+\sqrt{2(xy+yz+zx)}\right)\left(x+y+z-\sqrt{2(xy+yz+zx)}\right).$$ But ...
16
votes
1answer
437 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 ...
9
votes
2answers
8k views

Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime [duplicate]

I have proven that in $\mathbb F_{p^2}^*$ exists an element $\alpha$ with $\alpha^8 = 1$. Let $f(X) := X^4+1 \in \mathbb F_p[X]$. How can I prove that $f$ is reducible over $\mathbb F_p$? Has $f$...
89
votes
3answers
2k views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
17
votes
5answers
2k views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem I....
8
votes
4answers
3k views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = {f_1}^{...
8
votes
5answers
1k views

Proving irreducibility of $x^6-72$

I have the following question: Is there an easy way to prove that $x^6-72$ is irreducible over $\mathbb{Q}\ $? I am trying to avoid reducing mod p and then having to calculate with some things ...
11
votes
2answers
696 views

Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.

So I have seen the similar question and answers on here for $x^4 +1$, but I am having trouble extending anything there to this polynomial... I understand it is fairly trivial with Galois theory, but ...
5
votes
2answers
1k views

Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.

This is what I'm trying to prove: Let $f(x)\in\mathbb{Q}[x]$ and $\deg(f(x))>1$. Prove that $f(x)$ is irreducible in $\mathbb{Q}[x]$ iff its reciprocal polynomial $f^*(x)$ is irreducible in $\...
4
votes
2answers
1k views

Irreducibility criteria for polynomials with several variables.

Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime. If it is $K[x]$, then there are several methods which can be used to check whether a given ...
6
votes
2answers
2k views

Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial....
4
votes
1answer
835 views

Polynomial with a root modulo every prime but not in $\mathbb{Q}$. [duplicate]

I recently came across the following fact from this list of counterexamples: There are no polynomials of degree $< 5$ that have a root modulo every prime but no root in $\mathbb{Q}$. ...
3
votes
1answer
2k views

How many irreducible factors does $x^n-1$ have over finite field?

The polynomial $x^n-1$ is needed to be factorized into irreducibles over finite field $\mathrm{F}_q$. How many are them? I guess the question is about of number of cyclotomic cosets. Let $p$ be the ...
4
votes
2answers
1k views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$ [duplicate]

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
4
votes
6answers
1k views

$x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$

I have seen the thread Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$ but this didn't really have a full solution. Is it true that if it is reducible then it can be factored into a linear ...
3
votes
2answers
481 views

Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?

I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$. I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
16
votes
3answers
10k 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 help ...
14
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5answers
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Show that $x^4 + 8$ is irreducible over Z

Is there an easy way to show that $x^4+8$ is irreducible over $\mathbb Z$ without trying to write it as a product of polynomials of lower degrees?
6
votes
1answer
783 views

$x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field

Let $c$ be an element of a field $F$ of characteristic $p$ (prime). Then how to show that $x^p -x-c$ is irreducible over $F$ if it has no root in $F$? I was trying using contradiction and by ...
9
votes
3answers
1k views

The degree of $\sqrt{2} + \sqrt[3]{5}$ over $\mathbb Q$

I know that the degree is at most $6$, since $\sqrt{2} + \sqrt[3]{5} \in \mathbb Q(\sqrt{2}, \sqrt[3]{5})$, which has degree $6$ over $\mathbb Q$. I'm trying to construct a polynomial with root $\sqrt{...
5
votes
1answer
6k views

Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$

Determine the splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$ Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
16
votes
10answers
2k views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
12
votes
3answers
2k views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in $\...
7
votes
1answer
211 views

Irreducibility of an infinite sequence of polynomials

Prove that $P_n(X) = X^n - X^{n-1} - X^{n-2} - ... - X - 1$ is irreducible over $\mathbb{Z}$ for all $n$. I was able to prove the result for $n=2^k-1$ by applying Eisenstein's criterion to $P_n(X+1)$....
6
votes
2answers
192 views

Irreducibility of $X^n-a$

Let ${\mathbb K}$ be a subfield of ${\mathbb C}$. Let $a\in{\mathbb K}$ such that $X^d-a$ has no root in ${\mathbb K}$, for any divisor $d>1$ of $n$. Does it follow that $X^n-a$ is irreducible ...
5
votes
1answer
957 views

Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$

Is it possible to determine how many irreducible factors has $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? (Here $p,q$ are ...
5
votes
2answers
1k views

Find the polynomials which satisfy the condition $f(x)\mid f(x^2)$

I want find the polynomials which satisfy the condition $$f(x)\mid f(x^2).$$ I want to find such polynomials with integer coefficients, real number coefficients and complex number coefficients. ...
2
votes
4answers
700 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.