All Questions
19
questions
0
votes
2
answers
82
views
Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?
Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
- 15
0
votes
1
answer
77
views
Show that GF(81) is an $x^{26}+x^{8}+x^{2}+1$ decomposition field
I tried decomposing the polynomial, but after taking out $(x^{2}+1)$ you have to break the remainder into polynomials of degree 4, which is manually hard. Perhaps this is solved by using Frobenius ...
- 15
1
vote
0
answers
48
views
Interpolation of permutation polynomials
Consider the finite field $\mathbb F_q$ where $q=2^n$ and $n \to \infty$. Now given $t = O(1)$ and $x_1,\ldots,x_t,y_1,\ldots,y_t$ where $\forall i \ne j,x_i \ne x_j,y_i \ne y_j$, do one has a ...
- 21
3
votes
1
answer
210
views
Structure of multiplicative subgroup of a finite field
Consider a finite field $GF(q)$. We refer to $GF(q)^{\times }$ as the multiplicative group of $GF(q)$. Given that $\left| GF(q)^{\times }\right| =q-1$ and $GF(q)^{\times } \cong \mathbb{Z}_{q-1}$ ...
- 51
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
vote
1
answer
95
views
Proof of sub field and polynomial ring
My following task is:
Let $\mathbb{K}$ be a field und be $\mathbb{K}^{\prime}$ a subfield of $\mathbb{K} . \mathbb{K}[t]$ and $\mathbb{K}^{\prime}[t]$ are polynomial rings over the respective fields ...
- 53
7
votes
2
answers
100
views
When does the underlying map of a polynomial induce a permutation on $\mathbb{Z}/p\mathbb{Z}$?
For example, the underlying function of the polynomial
$$f(x)=4x^2-3x^7$$ induces a permutation on $\mathbb{Z}/11\mathbb{Z}$, though I only know the proof by "brutal force" (is there a cleverer proof?)...
- 5,451
1
vote
1
answer
417
views
How to find the root of the polynomial $x^2+x+1$ over $\mathbb{Z}_2$ in this field?
I am having some troubles understanding the proof for a statement. The question is:
Suppose R is the polynomial ring $\mathbb{Z}_2[x]$. Let $(x^4+x+1)$, I, be the principal ideal of this ring. ...
- 393
4
votes
1
answer
845
views
Roots of polynomial over finite field form a group
I was playing around with some polynomials over finite fields (specifically $\mathbb{F}_p^*$ where $p$ is prime), and I was wondering if there is in general a condition for the roots of a polynomial ...
- 1,527
2
votes
1
answer
681
views
For $p$ prime, is the polynomial $x^p-x+1$ irreducible in $\mathbb{Z}_p$? [duplicate]
It is possible, for $p\in\mathbb{N}$ prime, that the polynomial
$x^p-x+1$ is irreducible in $\mathbb{Z}_p$?
By the identity
$a^p\equiv a$ mod $p$ for any $a\in \mathbb{Z}_p$ surely there is not a ...
- 8,856
3
votes
0
answers
50
views
Order of $x$ in $\mathbb F_q[x]/(g(x))$
Let $\mathbb F_q$ be a finite field and $g(x)$ be a polynomial in $\mathbb F_q[x]$ with non zero constant term. Given the factorization of $g(x)=p_1(x)^{e_1}\dots p_k(x)^{e_k}$, give an estimate for ...
- 1,128
6
votes
0
answers
113
views
Factorization of certain polynomials over a finite field
Suppose we have the polynomial $$F_s(x)=cx^{q^s+1}+dx^{q^s}-ax-b \in \mathbb{F}_{q^n}[x]$$ where $ad-bc \neq 0$. The fact that $ad-bc \neq 0$ means that we can take the coefficients of $F_s(x)$ as ...
- 61
1
vote
3
answers
362
views
Cardinality of multiplicative group of a field $GF(q)/f(x)$ where $f(x)$ is an irreducible polynomial over GF(q)
If you have an irreducible polynomial $f(x)$ over $GF(q)$, then the following should be true:
$GF(q)/f(x) \cong GF(q^k)$, where $k = deg(f(x))$.
So the multiplicative group should have $\varphi(q^k)$ ...
- 133
0
votes
2
answers
360
views
Power of a polynomial in Galois field
Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true?
$$f^{p^m}(X) = f(X^{p^m}).$$
I tried to prove the assertion above and got stuck at the following:
$$
\begin{align}
f^{...
- 373
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