This tag is used for questions on polynomials rings in an arbitrary number of variables related to different topics studied in ring theory and commutative algebra. Questions related to high-school polynomials level or similar should use the tag "polynomials".
Questions tagged [polynomial-rings]
396 questions
2
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0 answers
Counting $(0,1)$-polynomials $f \in \mathbb Z/n\mathbb Z[x]$ that are multiples of $x(x+1)\cdots(x + n - 1)$.
I'm interested in generalizing OEIS sequence A329479:
Number of degree $d$ polynomials $f$ with all nonzero coefficients equal to $1$ such that $f(k)$ is divisible by $3$ for all integers $k$.
In particular, I'm looking for a way to compute…
Peter Kagey
- 5,052
1
vote
1 answer
Find a inverse of polynomial in quotient ring in $\mathbb{Z}_p$
Let $f=x^3+x+1$, I want to find a inverse of $g=x^4 +3x^3+x^2+1$ in $A=\mathbb{Z}_5[x]/(f)$.
I do the division between $f$ and $g$ is $q=x+3$ and $r=2x+3$, then I continued with the division between $x^3+x+1$ and $r$ and then I have rest $=1$.
Now I…
sickdamn
- 29
1
vote
2 answers
What are the coefficients of $x^2+2\in(\mathbb{Z}/\mathbb{Z}4)[x]?$
If $$(\mathbb{Z}/\mathbb{Z}4) = \{\bar{0}, \bar{1}, \bar{2}, \bar{3}\}$$
How can it be possible that $x^2+2 \in (\mathbb{Z}/\mathbb{Z}4)[x]$?
In other words, how can the 2 value in $x^2+2$ be expressed as a member of ($\mathbb{Z}/\mathbb{Z}4)[x]$ if…
ajf1000
- 111
0
votes
1 answer
Finding the inverse of $x^2-x-2 \in \Bbb{Z}_5[x]/(x^3-2x^2+2)$
I'm a relatively new student to this topic, please give comprehensive explanations
This is what I have so far:
We can use Euclidean algorithm to solve the following and we have:
$$\gcd(x^3-2x^2+2,…
Mengen Liu
- 109
0
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1 answer
How to prove that the order of $F[x]/(g(x))$ is $|F|^{\deg(g)}$, where $F$ is a finite field?
In the textbook that I am currently learning from, it says that:
Given $F[x]/g(x)$ for some fixed $g(x) \in F[x]$ of degree $d ≥ 1$,
$$F[x]/g(x) = {r(x) + g(x):\ \deg(r) ≤ d − 1}$$
If $F$ is finite, $r(x)$ has $d$ coefficients, each of which has…
0
votes
1 answer
What do the elements of a ring $R/I$ look like?
I am a bit confused, as my class is currently on Section 9.2 of Dummit and Foote, and what elements of $R/I$ are of the form of, where $R$ is a polynomial ring.
For example, I am reading that $\mathbb{Z}[x,y]/(x^2,y^2,2)$ are of the form $a + bx +…
Moni145
- 2,142
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ideals in the ring $\frac{F[x]}{(p(x))}$
Exhibit all the ideals in the ring $\frac{F[x]}{(p(x))}$, where $F$ is a field and $p(x)$ is a polynomial in $F[x]$ (describe them in terms of the factorization of $p(x)$).
Let $p={p_1}^{k_1} {p_2}^{k_2} ... {p_n}^{k_n}$ where $p_i\in F[x]$ is…
Vinay Deshpande
- 779
- 3
- 14