All Questions
Tagged with prime-factorization polynomials
39
questions
0
votes
1
answer
68
views
Reducible/Irreducible Polynomials in Ring Theory
I have this following exercise I've been trying to solve for a while now.
We are supposed to study the irreducibility of the polynomial $A=X^4 +1$ in $\mathbb{Z}[X]$ and in $\mathbb{Z}/p \mathbb{Z}$ ...
- 11
1
vote
2
answers
69
views
Factorization over $\mathbb{Q}$ and $\mathbb{Z_{41}}$
Factor $f(x) = x^4+1$ over $\mathbb{Q}$ and over $\mathbb{Z_{41}}$.
1)I can't factor $f(x)$ over $\mathbb{Q}$ because $f(x+1)$ is irreducible by Eisenstein's criterion.
2)I don't know where to start:
...
- 45
2
votes
1
answer
86
views
Show that for an odd integer $n ≥ 5$, $5^{n-1}\binom{n}{0}-5^{n-2}\binom{n}{1}+…+\binom{n}{n-1}$ is not a prime number.
I would prefer no total solutions and just a hint as to whether or not I’m at a dead end with my solution method.
So far this is my work:
From the binomial expansion,
$$\sum_{j=0}^n 5^{n-j}(-1)^j\...
- 183
4
votes
0
answers
243
views
Proper divisors of $P(x)$ congruent to 1 modulo $x$
Let $P(x) $ be a polynomial of degree $n\ge 4$ with integer coefficients and constant term equal to $1$. I am interested in Polynomials $P(x) $ such that for a fixed positive integer $b$, there are ...
- 244
1
vote
1
answer
170
views
Method to find if a polynomial has irrational roots?
Question
Let's say I define a polynomial $P(x)$ whose roots $\alpha_i$ and degree $> 1$.
We also add constraint that the coefficient of the highest power of $P(x)$ is $1$ and all other coefficients ...
- 1,187
2
votes
0
answers
92
views
Non-constant polynomial over an integral domain without any irreducible factors.
Let $R$ be an integral domain. I am trying to find a $f \in R[x]$, such that $\deg(f) \geq 1$, and $f$ does not have any irreducible factors in $R[x]$.
Does such $f$ exist?
Though I haven't been able ...
- 69
2
votes
3
answers
104
views
Show factorization $ x^{2n}-1=(x^2-1) \prod_{k=1}^{n-1}(x^2-2x \cos \frac{\pi k}{n} + 1) $
I'm interested in how to show that
$$
x^{2n}-1=(x^2-1) \prod_{k=1}^{n-1}(x^2-2x \cos \frac{\pi k}{n} + 1)
$$
I've seen this equality too often, but have no idea how to derive it. I've tried the ...
- 1,162
5
votes
1
answer
104
views
Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ (Homework) [closed]
Define $f(x) = x^6 + x^5 + 3x^4 +x^3 + 3x^2 + x + 1$. Find the largest prime factor of $f(19) + 1$ This problem is from a homework set of my class at source: Alphastar.academy. I believe there a ...
- 147
1
vote
1
answer
270
views
Using the remainder theorem to prove a quadratic is a factor of a polynomial
For example, if I have $P(x) = 3x^4 + 5x^3 -17x^2 -13x + 6$ then to show that $x^2 + x - 6$ is a factor I individually show that $x+3$ and $x-2$ are factors using the factor theorem (i.e. $P(-3) = 0$ ...
- 703
1
vote
0
answers
49
views
RSA Number Factorization - question regarding polynomial validity
I am interested in factorizing RSA numbers as a coding challange. I understand the basic principle that $2$ large primes are multiplied together to generate the specific RSA number, however I also see ...
- 13
2
votes
1
answer
50
views
Implications of representative of $p$-adic factor $g$ of $f$ dividing $f$ in $\mathbb{Z}[X]$
The problem is from this paper (click for pdf) by Mark van Hoeij.
Let $f \in \mathbb{Z}[X]$ be monic and squarefree.
Let $B$ be a Landau-Mignotte bound for $f$, i.e. for any rational factor $\phi$ ...
- 1,118
1
vote
1
answer
169
views
Degree $3$ polynomial with constant coefficient $2010$
$\mathbf{Statement}$: Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $|\alpha|>10$. (TRUE OR FALSE?).
...
- 2,620
3
votes
2
answers
284
views
When is $y=3x^2+3x+1$ a prime number in $\mathbb{Z}$ with $x \in \mathbb{Z}$?
The first few values of $y=3x^2+3x+1$ for integer values of $x$ are $7, 19, 37, 61, 91$, and $127$. I am wondering under what conditions of $x$ is $y$ a prime number?
I had initially hoped that ...
2
votes
2
answers
54
views
Units, Primes and Irreducibles
How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$?
Thank you.
- 51
2
votes
0
answers
180
views
Why are polynomials of even powers better for Pollard's rho?
Taking all $C(900,2)$ combinations of the first 900 prime numbers, I defined $N = pq$, where $p$ and $q$ are a combination of primes. Then I factored $N$ using Pollard's Rho, counting how many ...
- 145