All Questions
11
questions
4
votes
2
answers
228
views
Help with a finite field exercise. How to find the minimal polynomial of a given root in a given field.
I need a help with this exercise.
(i) Find a primitive root $\beta$ of $\mathbb{F}_2[x]/(x^4+x^3+x^2+x+1)$.
(ii) Find the minimal polynomial $q(x)$ in $\mathbb{F}_2[x]$ of $\beta$.
(iii) Show that $\...
- 391
5
votes
1
answer
224
views
Does there exist a polynomial indicator function over $\mathbb{Z}_p[x_1,\dots,x_{p^3}]$ of degree at most $O(p^2)$?
The Problem
Let $p$ be a prime. Does there exist a $p^3$-variable polynomial $P$ over $\mathbb{Z}_p[x_1,\dots,x_{p^3}]$ such that
$P(\boldsymbol{0}) \equiv 0 \ (p)$
$P(\boldsymbol{x}) \equiv 1 \ (p)$ ...
- 2,740
-1
votes
3
answers
376
views
Prove that $f(x)=(x^2-2)(x^2-3)(x^2-6)$ has a root in $\mathbb{F}_p$ for every $p$ prime number [duplicate]
This is the solution that my book gives ($\mathbb{F}_p$ is the finite field with $p$ elements):
We can assume that $p\neq 2,3$ because $0$ is a root of $f(x)$ in $\mathbb{F}_2$ and $\mathbb{F}_3$. ...
- 918
3
votes
1
answer
144
views
Expected number of monomials in a random function over finite fields
Let $f : \{1,2\}^n \rightarrow \mathbb{Z}_3$ be a function from the multiplicative subgroup of order $2$ of $\mathbb{Z}_3$ over $n$ variables ($\{1,2\}^n$) to $\mathbb{Z}_3$, such that each coordinate ...
- 75
-1
votes
1
answer
1k
views
What is the inverse of an element of polynomial ring over finite field?
Let's consider the polynomial ring $\mathbb{F}_q[x]$. How to find the inverse of an element of this ring. For example, If I'm working over $\mathbb{Z}_7[x]$, what is the inverse of $x^2+x+1$. This is ...
- 175
1
vote
2
answers
58
views
Show that $x$ has order $8$ in $\frac{\mathbb{F}_5[x]}{(x^2+2)}$
I know that because $x^2+2$ is irreducible in $\mathbb{F}_5[x]$, then the quotient ring is a field. I'm pretty sure that the order of this quotient ring is $5*5*5 -1 =124 = 2*2*31$, because we have $5$...
- 1,862
8
votes
1
answer
1k
views
Multiplicative group modulo polynomials
When working over $\mathbb{Z}$, it is well known what the structure of the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$ exactly is:
If $n=p$ for a prime $p$, then $(\mathbb{Z}/p\mathbb{Z})^...
- 2,517
2
votes
1
answer
187
views
Group of units of a non-integral quotient ring
I would to like to know which product of cyclic groups the group $A^\times$ of units of the quotient ring $$ A = \mathbb F_5[X] / ((X^2-2)^2) $$ is isomorphic to.
I know that $A$ is not an integral ...
- 29
2
votes
5
answers
784
views
Why do Z/7 have no cubic root of 2?
I was reading a textbook and came across the following line:
Now we prove there is no cube root of 2 in $Z/7$. By noting that $(Z/7)^\times$ is cyclic of order 6, it will have only two third powers,...
- 149
2
votes
0
answers
49
views
Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?
Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$.
Is ...
- 121
5
votes
3
answers
347
views
Roots of $x^2 + 2x + 2$
I'm trying to show that there are infinitely many values of $p$ such that $x^2 + 2x + 2$ has no roots over $\mathbb{F}_p$. Is this easily solvable? (I kind of came up with it myself so I don't know.)...
- 4,075