All Questions
Tagged with finite-fields ac.commutative-algebra
59
questions
3
votes
0
answers
303
views
When does composing polynomials reduce the degree?
Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...
- 71
2
votes
1
answer
324
views
Uncountable cardinals and Prufer $p$-groups
Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...
- 599
28
votes
1
answer
1k
views
Algebraic dependency over $\mathbb{F}_{2}$
Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
- 583
19
votes
3
answers
2k
views
Classification of rings satisfying $a^4=a$
We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
- 62.1k
18
votes
5
answers
7k
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 ...
- 17.6k
6
votes
2
answers
3k
views
Quotient of $Z[x_1,...,x_n]$ by a maximal ideal is a finite field [duplicate]
I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x_1,...,x_n$ are complex numbers and $I$ is a maximal ideal of $\mathbb{Z}[...
6
votes
1
answer
369
views
Polynomiality of functions over residue rings
Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...
- 411
0
votes
1
answer
605
views
Number of Minimal left ideals in the full matrix ring over a finite commutative local ring
Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user39204
3
votes
1
answer
4k
views
Order of an element in a finite field
Let $\mathbb F_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F_p[x]$ an irreducible polynomial, $E = \mathbb F_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F_p$, $\lambda\...
- 33
4
votes
3
answers
825
views
Field with cyclic product group
If a field has a cyclic multiplicative group, is it necessarily finite?
- 1,401
4
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: ...
- 1,631
2
votes
2
answers
361
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 21
4
votes
1
answer
880
views
Elements of trace zero in a field extension
Let $K=F_q$ and $F=F_{q^3}$, define the set A={$x \in F$ : $Tr_{F/K} (x)=0$}.
Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?
- 1,153
3
votes
1
answer
1k
views
Inverse for a permutation over GF(2)
Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous ...
