All Questions
17
questions
1
vote
1
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109
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Prove that $(X^2 + 1)^n + p$ is irreducible over $\mathbb{Q}[X]$
Let $p$ an odd prime number, congruent to $3$ mod $4$. Prove that the polynomial $f(x) = (X^2 + 1)^n + p$ is irreducible over the ring $\mathbb{Q}[X]$, regardless of the value of $n$ (natural number).
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12
votes
1
answer
723
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Finding polynomial without constant term that commutes with $f(x)=x^3+3x$
Consider the polynomial $f(x)=x^3+3x$ over $\mathbb{Z}$.
I am trying to find a polynomial $g(x)$ $(\neq f^{\circ n})$ of any degree (or series) without constant term which commutes with $f$ (or any ...
- 10.8k
1
vote
0
answers
42
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If $f(x)=ax^{2p}+bx^p+c\in\mathbb{F}_p[x]$, prove that $f'(x)=0$
If $R$ is a commutative ring, then the set of all polynomials with coefficients in $R$ is denoted by $R[x]$.
$\mathbb{I}_p[m]$ is the integers mod $m$.
When $p$ is a prime, we will usually denote the ...
- 366
1
vote
0
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395
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Proving $\mathbb{Z}_p[x]$ has Unique Factorization.
I'm self-learning some number theory. I'm trying to prove or disprove the following:
$$\mathbb{Z}_p[x]\text{ has a Unique Factorization Theorem}$$ (where $p$ is some rational prime)
Consider some $f \...
- 331
0
votes
2
answers
99
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Units in $A = \mathbb{Z}_3[x]/(x^2+1)$
Let $A = \Bbb{Z}_3[x]/(x^2+1)$, the quotient ring by the ideal $(x^2+1)$. Which ones are units?
I did this question in a very boring way, merely listing all the possibilities and check. I cannot find ...
- 873
0
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4
answers
80
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Let $n$ be a positive integer and $p=4n+1$, a prime number. Show that $(\frac{p-1}2)!$ is a root of $x^2+1=0$ in $\mathbb Z_p$ [duplicate]
I have to prove the following part, but I can not. Please help me.
Let $n$ be a positive integer and $p=4n+1$, a prime number.
Show that:
$((p-1)/2)!$ is a root of $x^2+1=0$ equation over $\mathbb{Z}...
- 11
2
votes
1
answer
99
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Give an example of a finite ring $R$ and polynomials $f, g\in R[X]$ such that the polynomial division of $f$ by $g$ is not unique
Give an example of a finite ring $R$ and polynomials $f, g\in R[X]$ such that the polynomial division of $f$ by $g$ is not unique.
I know that $f$ can be written as $\sum_na_nX^n$ and $g$ can be ...
- 1,080
0
votes
1
answer
142
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Integer valued polynomials in several variables
For simplicity this is about polynomials in just two variables.
Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials
$X^iY^j$ and therefore as a sum of polynomials $p_{ij}\...
- 13.9k
1
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0
answers
45
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Invertibility of polynomials [duplicate]
The following statement popped up on a review sheet for an exam I have coming up. I understand the statement but am unsure how to prove it. I'm also a little unsure about how invertibility works in ...
- 333
2
votes
1
answer
141
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Two questions on the Gaussian integers [duplicate]
I have two questions on the Gaussian integers.
Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$?
Conversely, does any element in $\mathbb{Q}(i)$ that ...
user356400
7
votes
0
answers
384
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When is the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)+1$ reducible in $\mathbb{Z}[x]$?
This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$..
(A) Find all positive integers $n$ and integers $a_1,a_2,\ldots,...
- 49.8k
1
vote
3
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69
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For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?
I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + [2]_nx^7+[...
user310769
6
votes
2
answers
641
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Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]
A polynomial problem from my old algebra textbook:
$f(x)\in\Bbb Z[x]$ with leading coefficient $1$, $\deg f(x)\ge 1$, and both $f(0)$ and $f(1)$ are odd numbers, prove:
$f(x)$ has no root within $...
- 13.7k
3
votes
2
answers
246
views
For which $p$ and $q$ polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $F_p[x]$?
It easy to prove that polynomials $x^q-1$ and $(x+1)^q-1$ are coprime in $\mathbb{Q}[x]$ if $(q,6)=1$, since they don't have a common zero in $\mathbb{C}$, this can be seen geometrically.
My question ...
- 31
2
votes
1
answer
342
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Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$
I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$.
I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
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