All Questions
308
questions
0
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17
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AES S-box as simple algebraic transformation
The next matrix represents the Rijndael S-box according to wikipedia and other sources
$$\begin{bmatrix}s_0\\s_1\\s_2\\s_3\\s_4\\s_5\\s_6\\s_7\end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 &...
- 463
1
vote
0
answers
29
views
Order of $\mathbb F _p [x] / (f)$.
I could use some help with the following exercise:
Find the number of reducible monic polynomials of degree $2$ over $\mathbb F_p$. Show this implies that for every prime $p$ there exists a field of ...
- 485
0
votes
0
answers
54
views
When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]
I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
- 17
0
votes
0
answers
37
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Degree of factors of the Artin–Schreier polynomial in $\mathbb{F}_q$. [duplicate]
Consider the field $\mathbb{F}_q$, where $q$ is a power of $p$, say $q=p^n$. Let $f=x^q-x-a\in\mathbb{F}_q[x]$, with $a\in\mathbb{F}_q$.
I'm trying to determine the degree of the irreducible factors ...
- 329
3
votes
0
answers
109
views
Roots of a irreducible polynomial are linearly independent over a finite field.
Question. When does a irreducible polynomial contains linearly independent roots over a finite field?
Motivation. For a finite cyclic Galois extension $E/F$, if $\alpha\in E$ generates a normal basis, ...
- 2,503
0
votes
1
answer
90
views
Why can't individual terms of a summation not cancel each other in the 2nd case?
Below is from a paper.
$F(.)$ is a low-degree multivariate polynomial over $\mathbb F$ in $s$ variables.
Checking if $\sum_{x \in \lbrace0,1\rbrace^s} F(x) = 0$ will not prove that that $F(x) = 0\...
- 486
3
votes
1
answer
153
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Characterization of irreducible polynomials over finite fields - alternative proof?
By accident I have found the following characterization of irreducible polynomials over $\mathbb{F}_p$.
Lemma. Let $g \in \mathbb{F}_p[T]$ be a monic polynomial of degree $m \geq 1$. Then, $g$ is ...
- 165k
0
votes
0
answers
54
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Reed-solomon coding: question on proof of minimum distance
In the "coefficient view" of Reed-Solomon encoding, the message is interpreted to be coefficients of a polynomial m(x). The code word is $c(x) = m(x)*g(x)$ where $g(x)$ is a generator ...
- 151
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
1
vote
0
answers
72
views
Is $\{x^3, x+b\}$ a generating set of $\mathrm{Sym}(\mathbb F_q)$?
Let $q=2^n$ where $n$ is a sufficiently large odd number. Consider the fintie field $\mathbb F_q$ and the symmetric group $\mathrm{Sym}(\mathbb F_q)$ over it.
I use $x^3$ to denote the permutation $x \...
- 21
0
votes
1
answer
155
views
Polynomials GCD over finite field [duplicate]
I'm trying to find the GCD of $x^3+2x^2+3x+4$ and $x+2$ over $\Bbb Z _5[x]$
I tried to use GCD euclidean algorithm and got the folowing:
$x^3+2x^2+3x+4 = (x^2+3)(x+2)+3$
$(x+2) = (2x)3+2$
$3=3\cdot2 + ...
- 158
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
-1
votes
3
answers
180
views
"Low degree Polynomials do not have too many roots" - what exactly does this mean?
I am watching this video from a playlist on Algebraic Coding theory - https://www.youtube.com/watch?v=XH7npgKN93k&list=PLkvhuSoxwjI_UudECvFYArvG0cLbFlzSr&index=16 - this particular video in ...
- 486
0
votes
0
answers
19
views
Explicit bivariate resultant example
I have two polynomials modulo a prime $p$ with a common root $(x_0,y_0)$ with $|x_0|,|y_0|<\sqrt{p}$ and there are no other roots in $\mathbb F_p$ or no other roots of this size in $\mathbb F_p$ (...
- 6,245
8
votes
2
answers
267
views
How to factor a polynomial quickly in $\mathbb{F}_5[x]$
I was doing an exercise in Brzezinski's Galois Theory Through Exercises and needed to factor the polynomial
$x^6+5x^2+x+1=x^6+x+1$ in $\mathbb{F}_5[x]$. Is there a quick way to do this?
I can see it ...
- 11k