1
vote
0answers
28 views

Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
0
votes
1answer
48 views

Set of Solutions of A Quadratic Equation with Coefficients in $\{0,1,\cdots , \ p-1\}$

I was just playing with quadratic equations and this interesting question came into my mind. Say I have a set of quadratic polynomials $S=\{f_{(b,c)}(x)=x^2+bx+c:b,c\in \{0,1,\cdots, p-1 \}\}$ where ...
8
votes
1answer
58 views

Existence of root of a polynomial over $\mathbb F_p$.

I came accross the following question and I can't find an easy proof of this fact : Let $p\geq 17$ be a prime number such that $p\equiv 1 \pmod 4$. Show that for any $z\in \mathbb ...
1
vote
0answers
44 views

root of binary matrix

There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication. Is there any way to find ...
2
votes
1answer
210 views

Formal Derivative and Multiple roots

I am currently really stuck on the following problem: Prove that if f(x) in Fp[x] and Df = 0 (where D : Fp[x] → Fp[x] is the formal derivative) then there exists g(x) in Fp[x] such that f(x) = g(x)^p ...
1
vote
1answer
37 views

Maximal Distinct Roots in $F_q$

Let $a\in F_q[x]$, and let $r(\cdot)$ denote the number of distinct roots over $F_q$. For any $i|q$, prove that $$ \max_{\deg(a)=1}r(x^i-a)=r(x^i-x) $$
2
votes
2answers
302 views

Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root ...
0
votes
3answers
242 views

Which field will contain all the roots of a polynomial over $GF(p)$

Given a a polynomial with coefficients in $GF(p)$ and degree $d$, will that polynomial always have $d$ roots in $GF(p^d)$? The text I am reading seems to be implying that this is true but I can't see ...