All Questions
26
questions
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Prove that the order of 2 modulo 2n+1 is odd
Let $n$ be a positive integer such that there exists a polynomial $f(x) \in \mathbb{F}_2[x]$ of degree n such that $f(x)\cdot x^n f(1/x) = 1+x+\cdots + x^{2n}\in \mathbb{F}_2[x]$. Prove that the order ...
- 1,225
3
votes
1
answer
94
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Forms of factors of values of $(x^p-1)/(x-1)$?
I am considering, for an odd prime $p,$ the polynomial $f_p(x)=(x^p-1)/(x-1),$ which is known to be irreducible. I am wondering whether the following claim is true: If for some integer $a>1$ and ...
- 7,423
1
vote
1
answer
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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0
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0
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36
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Minimal form of rational, integer and complex polynomials?
Do all rational polynomial e.g $\frac{a}{b}x^n + \frac{c}{d}x^{n-1} + ... +\frac{e}{f}x + \frac{f}{g} = 0$ have a integer polynomial representation?
My thinking is:
$\frac{a}{b}x^2 + \frac{c}{d}x + \...
1
vote
1
answer
75
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Is $g(x,y)= \frac{f(x^{2y+1},y)}{f(x,y)}$ always an integer?
This question is similar to this other question:
Let $$ f(x,y):= \frac{x^y -1}{x+(-1)^y}$$
and $$ g(x,y):= \frac{f(x^{2y+1},y)}{f(x,y)}.$$
Let $y\ge1$ be an integer. Show that $g(x,y)$ is a ...
- 3,716
14
votes
1
answer
420
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Is $x^n-\sum_{i=0}^{n-1}x^i$ irreducible in $\mathbb{Z}[x]$, for all $n$?
Let the sequence of polynomials $p_n$ from $\mathbb{Z}[x]$ be defined recursively as $$p_n(x)= xp_{n-1}(x)-1$$
with initial term $p_0(x)=1$.
Then $$p_n(x)= x^n-\sum_{i=0}^{n-1}x^i $$
Question 1: is it ...
- 3,716
3
votes
0
answers
87
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Showing the irreducibility over $\mathbb{Z}[X]$ of polynomials similar to the cyclotomic polynomials
This question follows this other question.
Let $y$ be a natural number, $x$ a variable and $$ f(x,y):= \frac{x^{2y}-1}{x+1}$$ and
$$ g(x,y):= \frac{f(x,y)^{2y+1}-1}{(f(x,y)-1)(xf(x,y)+1)}.$$
For a ...
- 3,716
0
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0
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34
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Meaning of irreducible polynomial that is factor of $P$
I am reading the following lemma about polynomials:
Suppose that $x$ is a root of a polynomial $P$, $\pmod p$. Then the
irreducible polynomial $(T - x)$ is a factor of $P$
I am not sure I understand ...
- 1,609
2
votes
1
answer
159
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Irreducible polynomial divisible by all primes
Does there exist an irreducible non-linear polynomial $P(x)\in\mathbb{Z}[x]$ such that for any prime number $q$ there exists $t\in\mathbb{N}$ such that $q|P(t)$ ?
Also (dis)proving whether there ...
- 1,528
-1
votes
1
answer
46
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Show that a polynomial is irreducible on $\mathbb{Q}$ [duplicate]
I would like to show that $P(X)=X^4-20X^2+16$ is irreducible on $\mathbb{Q}$, how to proceed ?
-1
votes
1
answer
93
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Applying Eisenstein's criterion to $x^3 + x^2 − 2x − 1$? [duplicate]
Is it possible to apply a shift (to the variable $x$) and Eisenstein's criterion to show that the polynomial $f(x) = x^3 + x^2 − 2x − 1$ is irreducible over the rationals?
- 3,696
1
vote
1
answer
193
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Monic polynomial irreducible modulo finitely many given primes
There are irreducible monic polynomials over $\mathbb{Z}$ that are reducible modulo every prime number $p$ (e.g. $x^4+1$). Given a finite non-empty set $S$ of primes is there a monic polynomial over $\...
user700841
3
votes
2
answers
1k
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What are the steps involved in finding the Greatest Common Divisor of two polynomials?
Ultimately I'm trying to define all the steps necessary to go from this toy quartic polynomial modulus:
$$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$
to:
$$x = 18, 19$$
One of the recommended ...
- 271
10
votes
1
answer
340
views
What are the factors of this quotient given by Fermat's Little Theorem?
$\forall a,b \in \mathbb{Z}, p\in \mathbb{P}$, let
$$F_p(a,b) = \frac{(a+b)^p-a^p-b^p}{p}$$
Note:
$F_3 = ab(a+b)$
$F_5 = ab(a+b)(a^2+ab+b^2)$
$F_7 = ab(a+b)(a^2+ab+b^2)^2$
According to data ...
- 2,218
7
votes
1
answer
248
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How can I construct polynomials with "small" coefficients generating a prime "late"?
Let $f(x)$ be a polynomial with degree $5$, integer coefficients and positive leading coefficient. Let $M$ be the maximum of the absolute values of the coefficients. Assume the smallest non-negative ...
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