All Questions
52
questions
2
votes
2
answers
85
views
In polynomial quotient rings over $\mathbb{F}_2$, how to use Chinese Remainder Theorem to solve equations?
Suppose I work over the polynomial quotient ring $\mathbb{F}_2[x,y]/\langle x^m+1,y^n+1\rangle$, and I want to solve for polynomials $s[x,y]$ that simultaneously solve the equations
\begin{equation}
a[...
- 133
0
votes
1
answer
43
views
$(\mathbb{F}[T] \ / \ P^2 \mathbb{F}[T])^*$ cyclic if and only if $\text{deg}(P)=1$ where $P$ is an irreducible polynomial over a finite field [duplicate]
I am asked to prove that if $\mathbb{F}$ is a finite field of size $p$ prime and $P\in \mathbb{F}[T]$ is an irreducible polynomial over $\mathbb{F}$ then $(\mathbb{F}[T] \ / \ P^2 \mathbb{F}[T])^*$ is ...
1
vote
1
answer
462
views
Irreducible polynomial in field of positive characteristic
Let $F$ be a field of characteristic $p > 0$ and $C$ be an element of $F$ that is not a $p$-power. For a positive integer $s$, show that $x^{p^s} − C$ is irreducible, and its splitting field is of ...
- 743
0
votes
2
answers
269
views
Incorrect result for extended euclidean algorithm for polynomials
I am trying to follow an algorithm but I cannot get the correct result. I don't know if the calculations are wrong (would be surprised, since I checked them carefully with an online SageMath engine), ...
- 239
0
votes
1
answer
104
views
Evaluating a Polynomial over a ring
When we have a polynomial over a ring, then the co-efficients of the polynomial belong to the ring i.e. they are all mod some number. For example, $f(x) = x^3 − 3x^2 + 2x$ over $\mathbb Z_7$
However, ...
- 486
3
votes
1
answer
106
views
Evaluate $\sum_{r\in R^*}^{}{r^2}$ and prove that $R$ is a field
I have some trouble with the following exercise:
Let $R$ be a finite commutative ring s.t. $|R| = k$. Assuming
$$ x^k-x=\prod_{r\in R}(x-r) \in R[x]$$
how can we prove that $R$ is a field and ...
user655132
1
vote
0
answers
42
views
If $f(x)=ax^{2p}+bx^p+c\in\mathbb{F}_p[x]$, prove that $f'(x)=0$
If $R$ is a commutative ring, then the set of all polynomials with coefficients in $R$ is denoted by $R[x]$.
$\mathbb{I}_p[m]$ is the integers mod $m$.
When $p$ is a prime, we will usually denote the ...
- 366
1
vote
1
answer
51
views
What are some examples of numbering systems for which it is easy to generalize permutation polynomials?
I'm writing an assignment on permutation polynomials (PP). I've already explored quite a few generalizations and characterization of PPs over $\mathbb{Z}_p$ for $p$ prime (and finite fields in general)...
- 87
2
votes
1
answer
147
views
Maximal ideals in $\mathbb{F}_q[x,y]$ [closed]
Let $p \in \mathbb{P}$ be a prime and suppose that an integer $e > 1$ is given such that the polynomial $s_e = 1 + x + \dots + x^{e-1}$ is irreducible in $\mathbb{F}_p[x]$.
My question is the ...
- 380
1
vote
1
answer
151
views
Monic irreducible reversible polynomial
This is Exercise 245 of the book "Fundamentals of Error-Correcting Codes" by W. C. Huffman and V. Pless, page 145.
Show that a monic irreducible reversible polynomial of degree greater
than 1 ...
- 23.1k
0
votes
1
answer
179
views
The number of units in the quotient ring $\Bbb Z_5[x]/(x^4-1)$
I am asked to find the number of units in the quotient ring $\Bbb Z_5[x]/(x^4-1)$, where $\Bbb Z_5$ is the finite field consisting of 5 elements. I know that this ring has $5^4$ elements (which is not ...
- 1,802
2
votes
1
answer
101
views
Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$
Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$.
I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
- 739
3
votes
1
answer
430
views
Irreducible Polynomials In $F_3$
Let us say I have some irreducible polynomials in $F_3$
$$p(x) = x^3 + 2x + 2$$
and
$$p(x) - 1 = x^3 + 2x + 1.$$
Now, using the power of Maple and Wolfram Alpha, we can check that
$$p(x^{13}) = x^{...
- 272
0
votes
2
answers
401
views
Is it true that no polynomial (with the exception of constant polynomial) have an inverse in $\mathbb Z/p\mathbb Z$ (where p is prime)?
I was solving the following exercise:
Find the inverse of $p(x) = 1 + x$ in $R[x]$ over $\mathbb Z/5\mathbb Z$ or show that it does not exist.
and finding that it does not exist because if there is ...
- 325
10
votes
1
answer
1k
views
Determine whether a polynomial is irreducible
Consider the polynomial $P=X^5-X-1\in\Bbb{F}_3[X]$. I want to show that $P$ is irreducible. We can easily check it has no roots, so the only way it could not be irreducible is by being a product of ...
- 660