All Questions
129
questions
0
votes
0
answers
16
views
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
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
0
votes
0
answers
37
views
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
2
votes
1
answer
64
views
Fields of characteristic $p$ where there exists an element $a$ that is not a p:th power.
Suppose $K$ is a field of characteristic $p$, but that $K \neq K^p$, that is, there exists an element $a \in K$ such that there is no $b \in K$ so that $b^p = a$. We want to prove that this implies ...
- 1,276
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
0
votes
1
answer
43
views
$(\mathbb{F}[T] \ / \ P^2 \mathbb{F}[T])^*$ cyclic if and only if $\text{deg}(P)=1$ where $P$ is an irreducible polynomial over a finite field [duplicate]
I am asked to prove that if $\mathbb{F}$ is a finite field of size $p$ prime and $P\in \mathbb{F}[T]$ is an irreducible polynomial over $\mathbb{F}$ then $(\mathbb{F}[T] \ / \ P^2 \mathbb{F}[T])^*$ is ...
2
votes
2
answers
183
views
Show that $2$ is a square in $\mathbb{F}_p$, using an element $\alpha$ obtained from an extension field as a root.
This problem asks me to show that $2$ is a square in $\mathbb{F}_p$ iff $p \equiv \pm 1$ mod $8$ through a number of steps, the first step is the following problem:
Let $p$ be an odd prime and let $\...
- 169
1
vote
1
answer
41
views
Reduced degrees of polynomials over a finite field
Theorems for polynomials over infinite fields are often modified for a finite field $\mathbb{F}_q$ by reducing the exponents of each monomial in the polynomial so that they are of the form $X^k$ where ...
- 563
4
votes
1
answer
78
views
When a square of a polynomial root is in $\mathbb{F}_p$
Let $p > 5$ be prime. If $1 + 5x^2 + 5x^4$ has a root in $w \in \mathbb{F}_{p^2}$, then does $-1 + x + x^2$ have a root in $\mathbb{F}_p$? If $w$ is a root of the former, then $5w^2 + 2$ is a root ...
- 91
0
votes
1
answer
405
views
Irreducible polynomial in integers modulo p
I am a completing a past paper question and I am undecided on what method to use here. The question is:
For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are:
(1) Check each $a\...
- 99
6
votes
1
answer
295
views
Polynomial with no zeros in ultraproduct of finite prime fields
I am interested in properties of ultraproducts of finite prime fields.
Let $I$ be a set, $\mathcal U$ a non-principal ultrafilter on $I$ and $(p_i)_{i \in I}$ a family of prime numbers. Let $F$ be the ...
- 1,780
1
vote
2
answers
383
views
Struggling with polynomial Long Division in $GF(256)$
I'm currently trying to build a QR Code generator, which has lead me to studying a bit of abstract algebra - more specifically, finite fields.
I've studied quite a bit on my own and I'm at the point ...
- 31
1
vote
1
answer
50
views
If $F$ is a field $a\in F$, if $(x-a)^n$ for $n\geq 2$ divides $r(x)$ then $r(a)=r^{\prime}(a)=0$
Statment:
If $F$ is a field $a\in F$, if $(x-a)^n$ for $n\geq 2$ divides $r(x)$ then $r(a)=r^{\prime}(a)=0$.
Proof: Induction over $n$
Case $n=2$
Let $F$ be a field and $a\in F$ since $(x-a)^2 |r(x)$...
- 165