All Questions
Tagged with polynomials irreducible-polynomials
1,520
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
101
votes
4
answers
4k
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 $(x+1)^n=s$ ...
- 1,193
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
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
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
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
50
votes
1
answer
2k
views
Is $ f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?
In this document on page $3$ I found an interesting polynomial:
$$f_n=\frac{(x+1)^n-(x^n+1)}{x}.$$
Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$.
...
- 12.9k
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
38
votes
1
answer
3k
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.
- 7,926
36
votes
2
answers
9k
views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ is ...
- 2,585
27
votes
1
answer
691
views
Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$
The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
- 48.5k
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
26
votes
3
answers
29k
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 ...
- 555
26
votes
1
answer
964
views
If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?
First, I ask my question and then I add some explanations:
Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers $...
- 23.9k
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