All Questions
Tagged with finite-fields polynomials
797
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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
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20
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How to understand 'the problem of determining the exact number of monomials in P(x) given by a black box is #P-Complete'
In this paper:https://dl.acm.org/doi/pdf/10.1145/62212.62241, what's given is $P(x)$ a (sparse) multivariate polynomial with real(or complex) coefficients.
The author claimed two things.
It is known ...
- 61
1
vote
1
answer
32
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Equivalent polynomials over a finite field
Disclamer. I'm not good at math, and the last time I did it in school was 10 years ago, so I'm writing everything in my own words.
Suppose we are working with polynomials in the space of remainders ...
- 13
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54
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Does there exists something like the BKK Theorem for polynomials over finite fields?
I'm trying to count the number of common $\mathbb{F}_q^\times$-zeros of some polynomials $f,g \in \mathbb{F}_q[x,y]$. I thought I had a solution using the BKK theorem but I reread the theorem ...
- 41
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1
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92
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Extended euclidian algorithm
I'm trying to understand how the matrix form of the extended euclidian algorithm for polynomials works for a BCH code with coefficients from $GF(2^4)$ in https://en.wikipedia.org/wiki/BCH_code
for ...
- 315
2
votes
2
answers
85
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In polynomial quotient rings over $\mathbb{F}_2$, how to use Chinese Remainder Theorem to solve equations?
Suppose I work over the polynomial quotient ring $\mathbb{F}_2[x,y]/\langle x^m+1,y^n+1\rangle$, and I want to solve for polynomials $s[x,y]$ that simultaneously solve the equations
\begin{equation}
a[...
- 133
1
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86
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Number of irreducible polynomials of degree at most n over a finite field
We know that the number $N(n,q)$ of irreducible polynomials of degree $n$
over the finite field $\mathbb{F}_q$
is given by Gauss’s formula
$$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$
The number ...
1
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0
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29
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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
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28
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What's the point of the local zeta function?
I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
- 584
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2
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82
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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
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1
answer
77
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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
3
votes
0
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67
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Is this connection between prime numbers, prime polynomials, and finite fields true?
I recently learned of the following connection between prime numbers and prime polynomials in the field of cardinality $2$. Namely, you take a natural number $n$, and use the digits of the base $2$ ...
- 21.4k
2
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1
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73
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Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?
I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
- 474
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54
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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
6
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2
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151
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Irreducible polynomial in $\Bbb{Z}_2[x]$
Suppose $2k + 1 \equiv 3 \mod 4$ in $\Bbb{Z}_{\geq 1}$.
Is the polynomial: $p_k(x) = x^{2k + 1} + x^{2k - 1} \dots + x + 1$ irreducible in $\Bbb{Z}_2[x]$?
I do not know whether it is true or not...
(...
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