All Questions
Tagged with polynomials irreducible-polynomials
279
questions
104
votes
8
answers
32k
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 ...
- 1,559
54
votes
3
answers
18k
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 ...
- 11.1k
26
votes
2
answers
6k
views
When $X^n-a$ 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 ...
- 1,155
59
votes
2
answers
11k
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 $x^...
- 593
68
votes
2
answers
33k
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!
- 4,215
17
votes
3
answers
20k
views
How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$? [duplicate]
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 $...
- 23.2k
15
votes
2
answers
7k
views
Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
I have a question, I think it concerns with field theory.
Why the polynomial $$x^{p^n}-x+1$$ is irreducible over ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?
Thanks in advance. It bothers me for ...
- 151
13
votes
4
answers
17k
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 $X+1$, ...
- 255
15
votes
8
answers
12k
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.
- 3,454
22
votes
5
answers
10k
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 ...
- 1,816
18
votes
3
answers
10k
views
Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q}$?
In the finite field of $q$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to be $X^{q^n}-X$. Why is this?
I understand that $q^n=\sum_{d\mid n}dm_d(q)$, ...
- 311
6
votes
2
answers
2k
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 ...
- 917
39
votes
2
answers
6k
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-...
- 4,845
24
votes
7
answers
9k
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 ...
- 2,030
57
votes
7
answers
30k
views
Methods to see if a polynomial is irreducible
Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then ...
user977