Finite fields are structures arising in abstract algebra. 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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Bounds for general character sums over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $\chi$ be the canonical additive character of $\mathbb{F}_q$, let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$, and ...
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Order of $\mathrm{GL}_n(\mathbb F_p)$ for $p$ prime [duplicate]

While proving some facts about matrix group operations on finite fields, I stumbled across the following question: What is the order of the group of invertible $n\times n$ matrices over a ...
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Cyclotomic extension of $\mathbb{F}_p((T))$

I feel very confused about why adding n-th roots of unity to $\mathbb{F}_p((T))$ would give $\mathbb{F}_{p^n}((T))$. (Is this true?)
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Splitting field of $X^6-7X^4+3X^2+3$ over $\mathbb Q$ and $\mathbb F_{13}$

I want to find the splitting field and the degree of the splitting field over $\mathbb Q$ and $\mathbb F_{13}$ for the polynomial $X^6-7X^4+3X^2+3$. Over $\mathbb Q$ the polynomial factors as ...
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General construction of $\operatorname{GF}(2^k)$

Is there a general method of constructing fields of the form $\operatorname{GF}(2^k)$? (preferably something that is easily manipulable by a computer.) I know that one can look for an irreducible ...
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Infinite algebraic extension of a finite field

I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following ...
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Subfield of a Field

I read that any field $F$ has a unique smallest subfield $F_0$ (Dummit and Foote : Exercise 7.5.3). Consider the field $F = F_p \times F_p$. $(F_p,0)$ is a sub-field of $F$ since it has a unit ...
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Why can a matrix whose kth power is I be diagonalized?

Say A is an n by n matrix over the complex numbers so that A raised to the kth power is the identity I. How do we show A can be diagonalized? Also, if alpha is an element of a field of characteristic ...
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24 views

Fixed point of field automorphism

Let $F$ be a finite field of order $p^n$ for some prime $p$ and positive integer $n$. This is well known that group of field automorphism of $F$ is cyclic and generate by the following: ...
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Finite field such that for every $a \in F$ , $x^2=a$ has a solution for $ x \in F$

Let $F$ be a finite field such that for every $a \in F$, the equation $x^2=a$ has a solution for $x \in F$ , then what can we say about the number of elements in $F$ and characteristic of $F$?
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Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
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compute homogeneous ideal of projective n-space over Galois field Fp

Let $n$ be a positive integer, and $P^n$ be the projective $n$-space over Galois field $F_p$. How can I compute the homogeneous ideal $I(P^n)$?
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Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
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Find the number of integers $r$ such that the polynomial $x^{r}-a$ has a linear factor over $\mathbb{F}_{p^{n}}$

If we have a finite field $\mathbb{F}_{p^{n}}$, how does one determine the number of integers $r$ in $\{0,1, \ldots, p^{n}-2 \}$ for which the equation: $x^{r}=a$ has a solution for every $a \in ...
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Calculating of genus of a curve

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is ...
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Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
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Polynomial roots and finite fields

QUESTION: Suppose you are given a polynomial with integer coefficients. And assume it only has complex roots(no real roots). Does it necessarily follow that in a finite field(say mod $p$) that it ...
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Is there a recurrence formula for finding primitive polynomials in order to construct $\mathbb{F}_{p^n}$?

(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.) Let $p$ be a prime and $n\geq 1$ an integer. The finite field ...
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projective space over finite fields

Let $A,B$ be sets non empty sets. Let say that if $p\in A$ then $p$ is said to be a point and if $l \in B$ then $l$ is said to be a line. Let $C$ be a set of the form $\{p,l \}$ with $p \in A$ and ...
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35 views

3 is Square in finite field

Let $K$ be a finite field with $p^2$ elements. Show that $3$ is square in $K$. I know that 3 is sum of two squares. Thanks.
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Show that the map $\varphi $ is an isomorphism.

We defined the Generalized Reed-Solomon codes the following way: $\alpha=(\alpha_0,\alpha_1,\ldots,\alpha_{n-1})\in \mathbb{F}_q$, distinct elements of the finite field $\mathbb{F}_q$, $n\leq q$, ...
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60 views

How to construct a field with exactly 125 elements

How to construct a field with exactly n elements in general? Is there any method to do so? And In case no such field exists, how do you determine that? Thanks in advance
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Polynomial over finite field

I'm currently reading these notes on the simplicity of $PSL_n(F)$. At page 5 it is used that there exists an element x in fields with 4 or more elements such that both: $x\neq 0$ $x^2-1\neq 0$ I ...
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Points on elliptic curve over finite field

Find the points on the elliptic curve $y^2 = x^3 + 2x + 2$ in $\mathbb F_{17}$. Do I have to guess a first point and then use an algorithm to spit out all other points?
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Quotient Ring and finite fields

How is a quotient ring $\mathbb Z/p^e\mathbb Z$ (where p is prime and $e>2$) different from a finite field $\mathbb F_{p^e}$? When they are both rings, have the same elements? I thought a finite ...
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Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

I want to prove, that $\Bbb F_p^\times = MRP(p)$. I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$ But I do not know how to continue this idea.
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Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
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elementary properties of cyclotomic polynomials

How can one rewrite $1+x^2+x^4+x^8+\cdots x^{2^n}$ as a product of cyclotomic polynomials? more general how can we express $1+x^p+\cdots+x^{p^n}$, where $p$ is a prime, in term of product of ...
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Arithmetic background of this RNG code

I am trying to figure out the mathematical background of the random number generation of an old video game. It does iterations where it updates a 33-bit state consisting of the variables z (32-bit) ...
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When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
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Is the Frobenius automorphism a polynomial?

Let $p$ be a prime and $\Bbb{Z}/\Bbb{Z}_p$ the field of integers mod $p$, and since $\Bbb{Z}/\Bbb{Z}_p$ is a field we have the ring of polynomials in $X$ with coefficients in $\Bbb{Z}/\Bbb{Z}_p$ ...
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On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
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Generators of group of “unitary” matrices over a finite field

This is about a group related to $U(n,q)$ and $SU(n,q)$. I know from multiple sources the generators for these groups, but $U(n,q)$ is defined to be the group of matrices $A$ such that $A^*JA = J$ ...
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How can I predict what numbers “work” without brute force?

I've been doing some research with LFSRs and I have found something I can't explain. I've worked on it for years but I'm finally opening up to public involvement because I can't stand not knowing. ...
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polynomial algebras and their coefficients in prime fields

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for $ n \in \mathbb {N} $ and $ m \in \mathbb {N}^n$, if $\mathbb{F}$ be field $GF(2)$ and $ X_1,...,X_n$ be $n$ pairwise ...
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How to partition a finite vector space into affine subspaces all of the same dimension

Given an $n$-dimension vector space $V$ over a finite field $\mathbb F_q$ and a natural number $d<n$, the goal is to write $V$ as disjoint union of $d$-dimensional affine subspaces $v_i+V_i$: $$V = ...
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Additive subgroups of the finite field GF($2^m$)

Consider the set $G=\left\{ {0,1,...,{2^m} - 1} \right\}$. The elements of this set can be viewed as the elements of GF($q=2^m$) with appropriate addition/multiplication operations. For example, GF(4) ...
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matrix rank over finite field

Let $M$ be a square matrix over finite field $\Bbb F_p$. Let $N$ be the matrix over $\Bbb F_p$ obtained replacing every non-zero entry of $M$ by $1$. Is $Rank(N)\leq Rank(M)^{f_p}$? for some constant ...
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GCD of polynomials in $\mathbb{F}_2[x]$

How can one show that $\gcd(x^4+x+1,x^4-x)=1$ in $\mathbb{F}_2[x]$? Is it because the polynomials do not share any irreducible monic polynomials i.e. $1, x, x+1$ as factors?
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Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...
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Help with problem of the book Algebra of T. Hungerford

This is a problem in “Algebra” by T. Hungerford: If $|K|=q$ and $f\in K[x]$ is irreducible, then $f$ divides $x^{q^n}-x$ if and only if $\deg f$ divides $n$. I found it difficult to solve ...
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Explanation of division/reduction in a binary Galois Field using bit-shifts

I've seen a lot of algorithms reducing the result of a multiplication in a Binary Field by using only bit-shifts and XOR. The number of positions to shift seems to be derived from the polynomial, but ...
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Product of random binary vector with random binary matrix in GF(2)

Suppose we have a binary vector $f$ with dimensions $1×l$ such that each entry in the vector is generated independently with propability $q$ of being $1$. And we have a binary matrix $G$ with ...
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Show that a matrix is not diagonalizable over a finite field

I have a matrix: $$\ \left[ {\begin{array}{cc} 2 & 2 \\ 1 & 2 \\ \end{array} } \right] $$ which I need to show that it cannot be diagonalized over the finite field $\mathbb{F_3}$. ...
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How to find all of generators in a finite fields

How can I find all generators of a finite field? For example in GF(2^3) and X^3 + x^2 + 1 as primitive polynomial. I don`t want all of solutions. I need some hint and help to solve this problem. ...
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Irreducible representations of inhomogeneous linear transformations over $\mathbb{F}_q$

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $G$ be the group of inhomogeneous linear transformations over $\mathbb{F}_q$, that is $x\mapsto ax+b$ for some $a\in \mathbb{F}_q^*$ and ...
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What does it mean for two polynomials to be the same in this fundamental field extension theorem?

I just read about the following "fundamental" theorem of field extensions which is stated as follows: Let $F$ be a field, and let $\alpha$ and $\beta$ be elements of field extensions $K/F$ and $L/F$. ...
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$Z_p$ has a additive inverse (for all $a\in Z_p$)

Let there be a finite field donated $Z_p:p\in Primes$. prove that for all $a\in Z_p$ there is additive inverse. Due to the properties of $Z_p$, $m*P=o_F: m\in Z$ Let assume $(p-a)$ additive inverse. ...
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Why $f(0)\neq 0$ where $f $ is a polynomial over the field $F_q$ and $deg(f)=m > 0$?

To construct the Residue class ring $F_q[x]/(f)$ having $q^m-1$ non-zero elements. Is it necessary for $f(0) \neq 0$? Why or why not? I have worked with different examples such as $x^3+x=f \in ...
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Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...