All Questions
39
questions
0
votes
0
answers
28
views
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
3
votes
0
answers
67
views
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
1
vote
1
answer
89
views
prove $f(x)$ and $g(x)$ have the same number of roots in $\mathbb{F}_{p}$
Suppose $p$ is a prime,and $f(x)=\sum\limits_{k=0}^{p-1}k!x^{k}$, $g(x)=\sum\limits_{k=0}^{p-1}\frac{1}{k!}x^{k}$,prove $f,g$ have the same number of roots in $\mathbb{F}_{p}$
I tried to consider the ...
- 171
9
votes
2
answers
261
views
$x^m-1 \nmid f(x)$ in $\mathbb{Z}/p\mathbb{Z}[x]$ where $f(x)=(x+1)((x+1)^{2m}+(x+1)^{m}+1)$
Problem: Suppose that $p$ is a prime. Suppose that there is $m \in \mathbb{N}$ such that $p=1+3m$. Define:
$$f(x)=(x+1)((x+1)^{2m}+(x+1)^{m}+1) \in \mathbb{Z}/p\mathbb{Z}[x]$$
I would like to prove ...
- 2,719
0
votes
0
answers
51
views
Can an irreducible polynomial over $\mathbb{F}_{q}[T]$ have multiple roots?
Let $\mathbb{F}_{q}$ be a finite field of order $q = p^{l}$ for some prime $p$ and $l \geq 1$ and consider the ring of polynomials $R = (\mathbb{F}_{q}[T])[x] $. Can an irreducible element $g(x)$ in $...
- 1,625
0
votes
1
answer
260
views
construct irreducible polynomials of degree 32 over $Z_2[x]$ [duplicate]
I'm learning finite fields behind Advance Encryption Standard. As far as I know, the irreducible polynomial used in AES is $x^8+x^4+x^3+x+1$. This is because AES s-box is based on bytes(8bits). Now I ...
- 11
1
vote
0
answers
69
views
Consequence of Chebotarev's density theorem
I'm studying Chatzidakis Notes about pseudo-finite fields. During a proof she states the following (by $\mathbb{F}_p$ I mean the finite field with $p$ elements for a prime $p$):
Let $f_1(x),\ldots,f_m(...
- 47
5
votes
1
answer
263
views
monic irreducible polynomial in $\mathbb Z[x]$ have a multiple root in $F_p$ over $F_p[x]$ for some prime $p$
Could you help me with the following problem which I cannot solve ?
Let $f\in \mathbb Z[x]$ be a monic irreducible polynomial with $deg(f)>1$.
Prove or disprove that there is always a prime $p$ ...
- 65
0
votes
0
answers
189
views
Chevalley–Warning theorem's proof
I'm struggling to prove Chevalley–Warning theorem, i.e. only the part which shows that
the number of common solutions ${\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}}$ is divisible by the ...
- 169
0
votes
2
answers
87
views
What is the cardinality of a vanishing set?
In Wiki's page on Chevalley–Warning theorem, under "Statement of the theorems", it's written that
Chevalley–Warning theorem states that [...] the cardinality of the
vanishing set of ${\...
- 169
6
votes
3
answers
504
views
Polynomial with roots modulo all primes $p \equiv 3 \pmod 4$
Does there exist an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $n \geq 2$ with a zero modulo all primes $p \equiv 3 \pmod 4$?
For example, there is such a polynomial $X^2+1$ if we choose ...
- 1,202
1
vote
1
answer
121
views
Summation problem with prime polynomials over finite field
Let $q$ be some prime number. Define, for $T<|\frac{1}{q}|$, $Z(T)=\sum_{f\in M_q}T^{deg(f)}$, where $M_q$ is the set of monic polynomials in $\mathbb{F}_q[x]$. Prove that $Z(T)=\prod_{p}\frac{1}{1-...
- 1,033
-1
votes
1
answer
1k
views
What is the inverse of an element of polynomial ring over finite field?
Let's consider the polynomial ring $\mathbb{F}_q[x]$. How to find the inverse of an element of this ring. For example, If I'm working over $\mathbb{Z}_7[x]$, what is the inverse of $x^2+x+1$. This is ...
- 175
4
votes
4
answers
829
views
Irreducible polynomials of degree greater than 4 over finite fields
I want to build a field with $p^{n}$ elements. I know that this can be done by finding a irreducible (on $Z_{p}$) polynomial f of degree n and the result would be the $Z_{p}$/f.
My question is ...
- 447
0
votes
1
answer
185
views
$L(s,\chi)=1$ for non trivial Dirichlet character modulo a polynomial of degree $1$
Let $m(x)\in \Bbb F_q[X]$ be a monic polynomial of degree $1$.
Show that for every non-trivial Dirichlet character modulo $m$ we have:
$L(s,\chi)=1$.
I have seen a theorem that states that if $\...
- 2,393