All Questions
11 questions
18
votes
5
answers
8k
views
Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
- 17.9k
15
votes
2
answers
1k
views
Can you use Chevalley‒Warning to prove existence of a solution?
Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...
- 28.7k
11
votes
1
answer
475
views
Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials
Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
- 63.1k
10
votes
1
answer
327
views
Proving that polynomials belonging to a certain family are reducible
In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.
Assume that $\mathbb F_p$ is ...
- 687
7
votes
1
answer
1k
views
Polynomials which are functionally equivalent over finite fields
Recall that two polynomials over a finite field are not necessarily considered equal, even if they evaluate to the same value at every point. For example, suppose $f(x) = x^2 + x + 1$ and $g(x) = 1$. ...
- 1,703
5
votes
1
answer
223
views
Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
- 6,224
5
votes
1
answer
2k
views
Irreducibility of some trinomials modulo $p$
Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: ...
- 1,641
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
4
votes
0
answers
191
views
Chevalley-Warning for finite rings: the degree of a non-polynomial
$\def\F{\mathbb F}$
$\def\Z{\mathbb Z}$
One reason that Chevalley-Warning theorem is that amazingly useful is the fact that for a finite field $\F$, any function from $\F^n$ to $\F$ is a polynomial. ...
- 23k
2
votes
0
answers
121
views
When $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))$?
Let's $P,Q\in\mathbb K[x]$, with $\mathbb K$ a finite field.
On what necessary and sufficient condition on $R \in \mathbb K[x]$ is it :
$\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))...
- 4,074
1
vote
1
answer
802
views
Trace 0 and Norm 1 elements in finite fields
Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...
- 255