# Questions tagged [polynomial-congruences]

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.

48 questions
Filter by
Sorted by
Tagged with
24 views

### What is the inverse of $[x^4+x^3+1]$ in $Z_{2}[x]/(x^2+x+1)$? [duplicate]

$x^4+x^3+1$ and $x^2+x+1$ are co-prime in $Z_{2}[x]$, so $[x^4+x^3+1]$ is a unit in $Z_2[x]/(x^2+x+1)$. Then, $[x^4+x^3+1]$ has an inverse, which I thought I could find by finding $a,b,c,d$ satisfying ...
1 vote
33 views

### The number of solutions to the congruence equation $P(x) \equiv 0 (\mathrm{mod}\ p^\alpha)$

Let $P(x)$ be a polynomial with integer coefficients, and $p$ be a large prime. I want to find the number of solutions to the congruence equation $$P(x)\equiv 0(\mathrm{mod}\ p^\alpha).$$ In my ...
66 views

### 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 ...
• 350
1 vote
53 views

### corollary of the partition congruence

I was going though the paper of Ramanujan entitled Some properties of p(n), the number of partitions of n (Proceedings of the Cambridge Philosophical Society, XIX, 1919, 207 – 210). He states he found ...
• 23
34 views

• 143
232 views

### $F[x]/(p(x))$ contains the roots of $p(x)$

The following theorem and exercise are from "Abstract Algebra, An Introduction, 3rd Edition, Thomas W. Hungerford" Corollary 4.19 Let $F$ be a field and let $f(x) \in F[x]$ be a polynomial ...
• 1,669
161 views

• 41
122 views

• 53
497 views

### 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 ...
1 vote
74 views

### Let $f(x)=x^3+x^2-5$. Show that for $n=1, 2,3, ...$ there is a unique $x_n$ modulo $7^n$ such that $f(x_n)\equiv 0\pmod{7^n}$.

My gut feeling for solving this problem is to use strong induction. Starting with the base case $n=1$ we can check each of the seven congruence classes and find that $x_1=2$ is the unique solution. ...
• 157
122 views

• 1,809
142 views

### How to do congruence-class arithmetic?

When working through this question: Write out the addition and multiplication tables for the congruence-class ring $F[x]/(p(x))$. $F=\mathbb{Z_2}$; $p(x)=x^{3}+x+1$. [Question #1 in section 5.2: ...
• 37
550 views

### 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 ...
329 views

• 1,216
642 views

### Solve $x^2 -3x+2 \equiv 0 \pmod 6$ [closed]

I need help understading how to solve this equation. $x^2-3x+2 \equiv 0 \pmod 6$ I'm having trouble with the said equation.
• 111
1 vote
107 views

### Congruence modulo ($2^{10}$)

Good evening, i’d like to discuss the following congruence which i’m stuck with, with you, hoping to find answers : Find the number of solution of $$x^5-16x\equiv 0 \mod 2^{10}$$ I think i have to ...
• 5,544
101 views

### Prove that $a^2,a^4,...,a^{p-1}$ are the quadratic residues modulo $p$.

Let $p$ be an odd prime $(p,a)=1$. Then, by Fermat's theorem, $a^{p-1}$ is congruent to $1$ (mod $p$). If $p-1$ is the smallest positive value of $e$ such that $a^e$ is congruent to $1$(mod $p$), ...
596 views

### 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 ...
1 vote
141 views

### Elementary Number Theory: Cubic Congruences

Solve $5x^3 -2x + 1 \equiv 0$ mod 243. We're using Hensel Lifting theorem to solve. I am trying to use the example we did in class but I am not following. I know we need to the dericative and find ...
• 289
65 views

### 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 ...
• 103
80 views

### How to find the multiplicity of a polynomial?

How can you find the multiplicity of a polynomial? I have to find all $n \in \mathbb{N}$ such that $9\mid n^4+n^3-2n^2+n+4$. The recommended method is to solve by congruence but I haven't been able ...
158 views

### 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 ...
• 1,765
### 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 ...