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.
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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
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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 ...
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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 ...
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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 ...
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Polynomial Congruence Equation
Struggling with this question: $15x^3 -6x^2 + 2x +26 \cong 0 \mod343$.
Here is what I have so far:
By Hensel's Lemma if we have a solution to $f(x) \cong 0\mod p$ we can find solution to $f(x) \cong 0\...
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if a homogeneous polynomial is the square of a polynomial modulo $x^2 + y^2 - 1$, is it also congruent to a sum of squares of homogeneous polynomials?
Let $f(x, y)$ be a homogeneous polynomial with real coefficients in $2$ determinates $x , y$. Suppose that
$$f(x, y) \equiv g(x, y)^2 \pmod{x^2 + y^2 - 1}$$
for some polynomial $g(x, y)$, where $g(x, ...
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$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 ...
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Regarding output for $p$-adic expansion on PARI/GP
The input sqrt$(2+O(7^{10}))$ on PARI/GP yields the output:
$ 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + O(7^{10}).$
Which is essentially the solution of the congruence $X^2 \...
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Find coefficient $c$ that makes a valid congruences
I've got a problem from book Integer, Polynomials, and Rings by Ronald S. Irving, page 227.
The question:
Is there a value of the coefficient $c$ in the field $\mathbb{R}$ that makes $x^4+3x^3+2x+1 \...
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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_{...
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Show that a congruence involving prime power is solvable.
Let $a,b$ be integers not divisible by a prime $p$, show that if $ax^p \equiv b\pmod {p^2}$ is solvable then, $ax^p \equiv b\pmod {p^n}$ is solvable.
What I've tried is by letting $x = w + v$, then $...
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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 ...
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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. ...
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Two polynomials congruence modulo $p$ [closed]
This is a result that I found in a compettion. But I didn't known it is false or true.
Let $n$ be a positive integer and $p$ be a prime divisor of $n$. Prove that if
$$x^{\varphi(n)}-1\equiv (x^{p-1} -...
user611437
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How to solve a non-linear system of modulus equations?
I have the following problem:
$$ 2x^2 + 8 \equiv 6 \;(\bmod\;13)$$
$$x \equiv 2 \;(\bmod\;15)$$
I have tried applying the Chinese remainder theorem, but could not figure out how to make it work, as ...
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