Questions about congruences where the modulus is a polynomial. For questions concerning congruences between polynomials where the modulus is an integer, use the tag (modular-arithmetic) instead.
Questions tagged [polynomial-congruences]
50 questions
6
votes
5 answers
Solutions of $x^2-6x-13 \equiv 0 \pmod{127}$
I started learning number theory, specifically polynomial congruences, and need help with the following exercise. Here it is:
Does the congruence $x^2-6x-13 \equiv 0 \pmod{127}$ has solutions?
I tried to follow the method for solving general…
user347616
4
votes
1 answer
How many quadratic functions mod 12 have exactly two roots?
There was a challenge question in this Socratica video and [EDIT: I misunderstood the question and] boy is it giving me a headache!
I thought the question was:
How many $(a,b,c)$ triples in $\Bbb Z_{12}$ satisfy $|r|=2$ where $r$ is the largest…
vaebnkehn
- 143
4
votes
1 answer
Find $f\in \mathbb{Z}_{p}\left[X\right]$ such that $f\left(X^{m}\right)$ is divisible by $\Phi_{p-1}$.
Let $p$ be a prime natural number, let $m\in\left\{ 2,\,\ldots,\,p-2\right\}$
and let $\Phi_{p-1}\in \mathbb{Z}_{p}\left[X\right]$ the cyclotomic polynomial corresponding to $p-1$. Find the polynomials $f\in \mathbb{Z}_{p}\left[X\right]$ such…
thebalans
- 103
3
votes
3 answers
Can I used polynomial congruence for prove 3 divide $n^3-n$?
I'm not sure this method can work to prove 3|$n^3-n$?
by let we have polynomial congruence
$n^3-n\equiv 0(mod3)$
then if all residue class mod 3 are the roots of congruence 3|$n^3-n$
the residue for mod 3 are 0,1,2
then I plug each of them in…
Lingnoi401
- 1,791
2
votes
1 answer
How to find all the roots in this ring?
Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$
Is it a field?
Find all the roots in F of the polynom $f (Y) := Y^2+[3]_{F}Y +[1]_{F} \in F[Y]$.
Attempt:
It is a field, because $x^2+3x+1$ is irreducible $\in \mathbb{F}_7[x]$. In fact it has no roots $\in…
Angelo Tricarico
- 199
2
votes
0 answers
Need hints on the following algebra problems.
I've been looking at these for over an hour and I don't understand how to do them. Any hints would be greatly appreciated.
Let $p(x) = x^3 + x + 1$ and $F = Z_3[x]/\langle p(x)\rangle$. Factor $p(x)$ in $F(x)$. Does it factor into linear…
Don Larynx
- 4,751
2
votes
1 answer
Is there any prime $p$ such that $6x^3 − p^2 − y^2 = 0$ has an integer solution?
I need to find whether there is any prime for which $6x^3 − p^2 − y^2 = 0$ has a integer solution.
For prime $p \neq 3$ ,considering this equation in modulo $3$ ,I find that there is no solution.
But for $3$ whether there is a solution or not I…
ビキ マンダル
- 388
2
votes
2 answers
Most efficient solution to find polynomial congruence for 0 mod p
I was given the polynomial $$f(x) = x^4 + 2x^3 + 3x^2 + x + 1$$ and told to find $$f(x) \mod 17 = 0 $$ I found the solution to be $$x = 8 + 17n$$ However, I arrived at this solution by computing all of the residues of f(x) mod 17 and then finding…
Nick Trotsky
- 87
2
votes
1 answer
Elementary Number Theory - Determining if there exist roots for a polynomial congruence with a prime modulus
If we consider something like the polynomial $f(x) = x^3-1$, and we want to know if there exists any solutions at all for $x^3 - 1 \equiv 0 \ (mod \ p)$, where $p$ is prime, is there a way to answer this without just plugging in every possible…
Stawbewwy
- 99
- 9
2
votes
2 answers
Lagrange Theorem Application in Number Theory
I'm studying the theory of congruences in number theory. I found a theorem called Lagrange Theorem that state
Given a prime p , let
$f(x)=c_0+c_1x + \cdots + c_n x^n$ be a polynonomial of degree n with integer coefficents such that $ c_n \not \equiv…
user135617
2
votes
0 answers
Let $R = \mathbb{Q}[x]$ and let $I = (x^2 + 2x + 2)R$ be the principal ideal generated by $x^2 + 2x +2$. Two questions are below.
i) Show that any element of $R$ is congruent modulo $I$ to a unique polynomial of the form $ax+b$ where $a,b \in \mathbb{Q}$?
ii) Show that any element of the quotient ring $R/I$ is of the form $\widehat{ax+b}$ for some $a,b \in \mathbb{Q}$?
I've…
M.Byrne
- 181
2
votes
1 answer
How to solve system of congruences of polynomial?
Find a polynomial $p(x)$ such that
$p(x)\ \equiv 1\mod\ x^{100}$ and
$p(x)\ \equiv 2\mod\ (x-2)^3$
ILiveInValhalla
- 103
2
votes
1 answer
solve x for a cubic congruence equation with large prime mod.
For $x^3 = 123456789 \pmod{1000000007}$ given $1000000007$ is a prime. Find $x$.
My school only teach us about linear congruence equation, and it is an extra credit question. Therefore, I think the question can solve by using the concept only in…
Hugo
- 33
2
votes
1 answer
Equality of polynomial functions modulo n
Fix positive integers $m$ and $n$. For all polynomial functions $f,g: \mathbb{Z}^m \to \mathbb{Z}$ define the equivalence relation $\sim$ by
$$f \sim g \iff \forall x \in \mathbb{Z}^m \ ( \ f(x) \equiv g(x) \pmod{n} \ )$$
$\sim$ is compatible with…
DAS
- 732
2
votes
1 answer
Understanding the following lemma
While studying primitive roots, I came across the following lemma:
Lemma: Let $p$ and $q$ be primes and suppose that $q^\alpha\mid p-1$, where $\alpha\geq 1$. Then there are precisely $q^\alpha - q^{\alpha -1}$ residue classes $a\pmod p$ of order…
Apurv
- 3,401