# Questions tagged [finite-fields]

Finite fields are fields (number systems with addition, subtraction, multiplication, and division) with only finitely many elements. They arise in abstract algebra, number theory, and cryptography. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.

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### Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
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### Why isn't the zero ring the field with one element?

I've heard that in the study of finite fields, and other concepts related to finite fields, mathematicians have found a sort of gap: there are various results and things that seem like they correspond ...
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### Irreducible polynomial roots and representations for Galois field elements in normal basis

I am trying to understand some topics in the literature and came across the following problem. Say I have a field $GF(2^4)$ defined by the irreducible polynomial $r(z) = z^4 + z + 1$, and I want to ...
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### Is $(A+B)$ necessarily singular?

Let $A, B$ be two orthogonal matrices over a field $F$ of characteristic $2$ such that $$\det (A) + \det (B) = 0.$$ Is $(A+B)$ necessarily a singular matrix? I have proved the result to be true for ...
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### Mean and variance of rank of a random matrix over finite field

I am trying to understand the statistics of the rank of a random matrix over a finite field, and some search brought me this paper. In Corollary 2.2 the paper gives the number of $k\times n$ matrices ...
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### A property of finite fields [closed]

Why does the remark in Wiki's proof of Warning's theorem true? If $i<q-1$ then $\sum _{x\in \mathbb {F} }x^{i}=0$ Best regards
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### Rank of matrix over $GF(2)$ whose rows have exactly $k$ elements $1$ [closed]

Consider the $\binom{n}{k}\times n$ matrix $A$ whose rows have $k$ $1$'s and $n-k$ $0$'s. There are no repeated rows. What is the rank of $A$ over $GF(2)$?
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### A quadratic solution and its conjugate

Let $\mathbb{F}_q$ be a finite field of $q$ elements such that $q=p^{t}$. Let $\alpha$ be a solution of the equation $Ax^2 +Bx +C=0$ and $\alpha'$ its conjugate, where $A$, $B$ and $C$ are nonzero ...
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### Elliptic Curve r-torsion points: Trace and AntiTrace

I am reading Pairings for Beginners from Craig Costello (pdf available for free, just google it) on page 53-55, We consider an elliptic curve E defined on a finite field $F_q$ with q prime q. We are ...
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### $A\in \mathbb{F}_q$, then there exists $n\in \mathbb{Z}_{\geq 0}$ such that $A \in \mathbb{F}_{q^n}$ a perfect square.

I was wondering whether the statement in my title holds. I think it does, but I am not sure. I have managed to prove it for the case $A = -3$, but not in the general case. Any ideas or tips?
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### Question about some term in Sage while using GF(9)

I tried to define an elliptic curve over $GF(9)$ in Sage, and some term $z2$ appeared, see below (click on the image if the font is too small): I know that it has something to do with the definition ...
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### Does $\Phi_n(\alpha)=0$ in $\Bbb{F}_p$ for some $\alpha\in\mathbb{F}_p$ imply that $\mathrm{ord}(\alpha) = n$?

Let $\Phi_n(x)$ denote the $n^\text{th}$ cyclotomic polynomial. Suppose it has a root $\alpha$ in the finite field $\Bbb{F}_p$ and $p \nmid n$. Does it follow that $\mathrm{ord}(\alpha) = n$? In the ...
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### Determine whether a polynomial is irreducible

Consider the polynomial $P=X^5-X-1\in\Bbb{F}_3[X]$. I want to show that $P$ is irreducible. We can easily check it has no roots, so the only way it could not be irreducible is by being a product of ...
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### Every Galois field $F$ of characteristic $p$ is perfect

I'm trying to do Exercise 2.6.13 from textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell. Could you please confirm if my attempt is fine or ...
Input: p (prime number), n (positive number) Output: g ( generator ) I have just found an irreducible polynomial over $F_p[x]$. Now I must find all generators of the multiplicative group from this ...