All Questions
21
questions
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$ p(x) $ as a product of irreducible (over $ \mathbb{F}$) factors.
Hello can you help me with this problem?
Let $ p(x) = x^5 + x^4 + x^3 + 2x + 2 \in \mathbb{Z}_5 [x].$ Construct the field $\mathbb{F} = \mathbb{Z}_5 [x] \setminus \langle x^3 + 2x + 1 \rangle $ ...
- 383
3
votes
1
answer
227
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When are polynomial functions over finite fields linear?
If I have a polynomial function $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ such that $f(x) = x^n$, under what circumstances will $f$ be linear? I think $f$ can only be linear if $n = p$, as ...
user555558
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0
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42
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Developed form of composition of polynomials in a finite field
Let $q=p^k$ be a prime power. Any polynomial $a(x)$ with coefficients in the finite field $\text{GF}(q)$ can be assimilated to a vector $a$ of $q-1$ elements $a_i\in\text{GF}(q)$, with (using the ...
- 1,768
2
votes
2
answers
99
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Linear Algebra (linearly independent)
I want to ask something related to exercise number 5 subchapter 1.5 in linear algebra book by Stephen Friedberg.
The question is, "prove that $\{1, x, x^2, \ldots, x^n\}$ is linearly independent in $...
2
votes
0
answers
50
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Proving the mean zero condition for polynomials over finite fields by linear algebra
Let $q:=p^n$ with $p$ a prime and let $\mathbb{F}_q$ denote a finite field with $q$ elements. Consider the following result:
If $f\in\mathbb{F}_q[X]$ has $\deg f\leq q-2$ then $$\sum_{x\in \mathbb{F}...
- 4,886
1
vote
0
answers
43
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Counting the powers of a companion matrix that possess nonzero leading principal minors
Let $p$ be prime, let $A$ be the $n\times n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in ${\mathbb Z}_p$, and let $T=\{A^j\mid 1\leq j\leq p^n-1\}\...
- 11
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69
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What is an efficient way of evaluating a polynomial when its roots are known?
Hypothesis: All calculations and polynomials are defined over a finite field of prime order.
Assume we know all roots $r_i$ of a polynomial: $P(x)=(x-r_1)(x-r_2)...(x-r_n)$ and a set of x-...
- 21
8
votes
1
answer
1k
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Roots of random polynomials.
Assume $P(x)$ is a random polynomial of degree $d$, where its coefficients are picked uniformly at random from $\mathbb{F}_p$, and $p$ is a large prime number. So the polynomial is defined over $\...
- 333
5
votes
1
answer
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Rank of a Polynomial function over Finite Fields
Consider a function $f:\mathbb{R}^k \to \mathbb{R}^n$ where $f=(f_1,f_2,\dots,f_n)$ with each $f_i$ being a polynomial function of variables $t_1,t_2,\dots,t_k$.
Now if the polynomials $f_1,f_2,\dots,...
- 2,517
0
votes
0
answers
89
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How to represent elements of a field by a base with minimal coefficients?
Let $K:=\mathbb F_{p^{16}}=\mathbb F[x]/(x^{16}-2)$ be a finite field where $p$ is chosen, that this ring-extension is really a field. Now we take an element $A\in K$ and we want to compute
$$A^d \in \...
- 317
1
vote
1
answer
414
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Dimension of the vector space of polynomials vanishing on a finite set
So, in
Size of vector space of polynomial that vanish on some set
the following question is asked. However, I am not convinced by the logic of the answer. Rather than revive an old question, I ...
- 1,372
6
votes
0
answers
113
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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
3
votes
1
answer
147
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Properties of polynomial in finite field
Let $f \in F_p[x]$.
Prove that polynomials $f(x), f(x + 1), \dotso, f(x + p - 1)$ are either pairwise distinct or they all coincide.
I think the right approach may involve Lagrange polynomials.
- 1,189
3
votes
1
answer
790
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What is the relationship between the generator matrix and polynomial of a FEC block code?
When talking about FEC (forward error correction) block codes, some literature uses matrix terminology and some talks about polynomials. I know that the same block code could be expressed with either ...
- 143
1
vote
2
answers
138
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Given two polynomials $p_1,p_2\in\Bbb{F}_p[x]$, how to define polynomial $p_3$ that has all roots of $p_1$ except those that are $p_2$'s roots?
Hypothesis: all polynomials are define over field $\mathbb{F}_p$, where $p$ is a large prime number.
Consider we have two polynomials, $p_1(x)$ and $p_2(x)$ (as defined above).
For simplicity ...
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