All Questions
Tagged with finite-fields ac.commutative-algebra
59
questions
0
votes
0
answers
96
views
Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
- 181
0
votes
0
answers
112
views
Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 181
2
votes
1
answer
200
views
Minimality implies algebraic independence?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 181
1
vote
0
answers
82
views
When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 181
0
votes
0
answers
177
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 913
8
votes
1
answer
617
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 181
0
votes
0
answers
116
views
A question on a system of quadratic polynomials
Consider the following system of quadratic polynomials $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,....,x_n]$ :
$f_1 (\bar{x}) = x_1 + x_n^2 + q_1$
$f_i(\bar{x}) = x_i + q_i$ for $i \in \{2,...,n-1 \}$
$...
- 181
0
votes
0
answers
109
views
Characterisation of even characteristic quadratic system
$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that
$f_1(\bar{x}) = x_1 + x_5^2 + q_1$,
$f_2(\bar{x}) = x_2 + x_1^2 + q_2$,
$f_3(\bar{x}) = ...
- 181
1
vote
1
answer
111
views
A question on classification of quadratic polynomials in even characteristic
$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...
- 181
1
vote
1
answer
98
views
Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
- 913
4
votes
1
answer
330
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 > ...
- 62.1k
0
votes
0
answers
89
views
Representing an $m$ dimensional quadratic polynomial as a polynomial on $\mathbb F_{q^m}$
We can represent $\mathbb{F}_{q^m}$ as $\mathbb{F}_q[\alpha]$ where $\alpha$ is root of an irreducible $m$-degree polynomial on $\mathbb{F}_q$.
By sending $\sum_{i=0}^{m-1} c_i\alpha^i \mapsto (c_{m-1}...
- 1
2
votes
0
answers
116
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))...
- 3,824
10
votes
1
answer
320
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
979
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
21
votes
1
answer
577
views
Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors
For all primes up to $p=89$ there exists a product $Q=\prod_{j=1}^d(x-a_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a_j$ in $\mathbb F_p[x]$ such that $Q'$ has all its roots in $\mathbb ...
- 17.6k
1
vote
2
answers
255
views
Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$
I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its ...
- 145
6
votes
1
answer
497
views
Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
11
votes
4
answers
1k
views
Explicit large finite fields in characteristic $2$
Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is ...
- 7,682
10
votes
1
answer
595
views
Discrete logarithm for polynomials
Let $p$ be a fixed small prime (I'm particularly interested in $p = 2$), and let $Q, R \in \mathbb{F}_p[X]$ be polynomials.
Consider the problem of determining the set of $n \in \mathbb{N}$ such that $...
- 12.3k
4
votes
1
answer
647
views
Jacobian criterion for algebraic independence over a perfect field in positive characteristics
It is well known that the Jacobian criterion for algebraic independence does not hold in general for fields of positive characteristics. However, the following partial statement seems promising:
...
- 935
0
votes
0
answers
82
views
Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
- 1,703
5
votes
1
answer
217
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 ...
- 6,162
2
votes
0
answers
66
views
Partitioning a given set into root set of two polynomials or non zero set of two polynomials simultaneously
Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...
- 850
11
votes
1
answer
472
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]...
- 62.1k
3
votes
1
answer
852
views
The relationship between a finite field and a quotient ring in $\mathbb{F}_p[x]$
Let $ f$ be an irreducible polynomial of degree $q$
over $\mathbb{F}_p$.
Let ${\bf F}=\frac{\mathbb{F}_p[x]}{f}$ be the finite field which contain $p^q$ elements.
Assume $k>1$ is an integer and ...
- 211
1
vote
1
answer
85
views
Converging sequence of polynomials
Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$ of degree $2$. Does there exist two polyomials $\alpha,\beta\in\mathbb F_q[T]$ (not both zeroes) such that the sequence $(\beta T^{q^{2n}}-\...
- 3,811
6
votes
0
answers
114
views
A question about the span of a sequence of polynomials satisfying a linear recurrence
Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...
- 5,412
1
vote
0
answers
1k
views
Are the integers a vector space or algebra over "some" field or over "some" ring?
Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
- 171
4
votes
0
answers
775
views
When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...
- 1,149