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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Is Z3 a sub-field of R?

The inverse numbers for the items in $\mathbb{Z3}$ are different than in $\mathbb{R}$ so I assume it's not a sub-field of $\mathbb{R}$. Am I correct? And in general, can sub-field of a infinite ...
3
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3answers
47 views

To find a field of $p^{2}$ elements ,where $p$ is prime

Show that there exists a finite field of $p^{2}$ elements for every prime $p\in\mathbb{N}$. What I thought is that if I find some irreducible polynomial of degree two over $\mathbb{Z_p[x]}$, then I ...
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0answers
31 views

Proving that one has solved chess by exhibiting the zeroes of polynomials over finite fields?

My question is based on one of Scott Aaronson blog post which states that a God-like being could convinced the villagers, to any degree of confidence, that she has solved chess by answering a few ...
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17 views

BCH Coding: Parity Check Matrix

I was referring to a publication on BCH codes in GF(64) (http://ipnpr.jpl.nasa.gov/progress_report2/42-38/38O.PDF) where I noticed that the Parity Check Matrix (attached image) that they have derived ...
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19 views

Number of non zero solutions to an equation in a field. [duplicate]

Let $F$ be a finite field of order $32$. Then the number of non-zero solutions $(a,b)$ belonging to $F*F$ such that $x^2+xy+y^2=0$ is equal to?? Please answer someone as i have no clue whatsever ...
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1answer
46 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
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2answers
41 views

Examples of fields with characteristic $2$. [on hold]

What are good examples of fields of characteristic $2$, starting from the simplest one to more interesting examples?
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1answer
63 views

Number of solutions in a field of order $32$ [duplicate]

Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$. As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, ...
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2answers
23 views

Existence of elements of even order in a field with characteristic 2

I've read this statement in a presentation slide, but it isn't obvious to me on why this is true: Forgetting about BCH codes, the question is: if an alement $\beta$ has even order ($2k$ is always ...
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1answer
12 views

order of a subrgoup of rank $r\geq 2$ in $\mathbb{F}_p^*$

Let $a,b\in \mathbb{F}_p^*$ with orders $o_p(a)=|\langle a \rangle|=\alpha$ and $o_p(b)=|\langle b \rangle|=\beta$. I have few questions: 1) Is it true in this case ($\mathbb{F}_p^*$ cyclic) that ...
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24 views

Twisted centralizer

Let $F$ denote a finite field and $A$ a square matrix with coefficients in $F$. The set of all matrices $B$ such that $BA=AB$ is called the centraliser of $A$. Now consider the set $C(a,A)$ of all ...
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28 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring [on hold]

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
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2answers
42 views

Number of solutions for the given equation in finite field of order 32.

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$. I have figured out that non zero elements of this field forms a cyclic multiplicative group of ...
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1answer
38 views

Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in ...
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1answer
36 views

$a^{(p-1)/n}=1$ implies $b^n=a$ for some $b\in\mathbb F_p$? [closed]

Let $p$ be a prime. Let $n$ be a positive integer dividing $p-1$. Suppose that $a^{(p-1)/n}=1$ in the finite field $\mathbb F_p$ with exactly $p$ elements. Then does there exist $b$ with ...
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1answer
27 views

A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
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0answers
20 views

Find GCD of polynomials over GF(101)

Hello all I'm teaching myself cryptography, and I'm struggling with polynomial arithmetic over finite fields. I've some what been able to teach myself how to do the arithmetic over $GF(2)$, but when ...
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0answers
15 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
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0answers
15 views

Minimize the rank of a matrix with some entries known

Let $m,n$ be two positive integers, with $m\geq n$. Suppose we have $m$ sets $A_1,\ldots, A_m\subseteq [n]$, with $|A_i|=d_i$. Let $\mathbb F$ be a finite field of size $q$. Let $D$ be the set ...
2
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0answers
21 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
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1answer
44 views

The number of elements in the special linear group over the finite field $\mathbb{Z}/p$ [closed]

I have $SL_{2}\{\mathbb{Z}/p\}$ for $p$ prime and $\mathbb{Z}$ integers. How do I show that this is a subgroup of $GL_{2}\{\mathbb{Z}/p\}$ and find the number of elements in $SL_{2}\{\mathbb{Z}/p\}$?
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37 views

On analogy between $\Bbb Z$ and $\Bbb F_q[x]$

There are objects and operations analogous between $\Bbb Z$ and $\Bbb F_q[x]$. For example primes in $\Bbb Z$ and irreducibles in $\Bbb F_q[x]$ are analogous and so is multiplication operation. ...
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1answer
47 views

A property of bijective polynomials

Let $F$ be a finite field and let $f\in F[x]$ be a non-linear polynomial such that it is bijective when considered as a function on $F$. Is it possible that the degree of $f$ divides $|F| -1$ ?
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16 views

Linear complexity of powers of a periodic sequences over finite fields

Let $\mathbf{a}^N = a_0 a_1\cdots a_{N-1}$ and let $\mathbf{a} = \mathbf{a}^N\mathbf{a}^N \cdots $ be an $N$-periodic sequence over the finite field $\mathbb{F}_q$ with $q$ elements, where $N \mid ...
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1answer
24 views

The algebraic structure of repeated root cyclic codes

Is there a generator theory for repeated root cyclic codes over finite fields? In other words is the ring $GF(q)[x]/(x^n-1)$ principal when $(n,q)>1$?
3
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18 views

Twisted Kloosterman Sums

A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of ...
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1answer
39 views

Multiplicative Inverse in a $256$ Galois Field

I am working on finding the multiplicative reverse in $GF(2^8)$ using the Euclidean Algorithm but after reading multiple sources, I feel as though I am proceeding incorrectly. Using the irreducible ...
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2answers
17 views

Character Sums and Characteristic Functions

Let $\mathbb{F}_q$ be the finite field with $q$ elements with $q$ odd. Consider the subgroup $H$ of $(\mathbb{F}_q)^{\times}$ consisting of squares: $H=\{x^2\,:\,x\in(\mathbb{F}_q)^{\times}\}$. We can ...
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63 views

Residue fields of schemes of finite type (over $\mathbb{Z}$)

Suppose $X$ a scheme of finite type over $\mathbb Z$. I want to prove that: (1) The residue fields of closed points of $X$ are finite; (2) For a given $q=p^n$ with $p$ prime, there is only a finite ...
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0answers
23 views

Existence of a linear map in general position

Let $W$ and $T$ be linear spaces over $\mathbb F$, with $n=\dim(W)$ and $t=\dim(T)$. Let $U_1,\ldots,U_m$ be a family of subspaces of $W$ with $\dim(U_i)=r_i$ for each $i$. The linear map $\phi:W\to ...
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1answer
22 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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1answer
43 views

field properties from prime subfield

Given a field $F_{p^n}$ with $char(F)=p$ and $p$ prime. And thus our main misunderstanding: Why is the arithmetic in $F_{p^n}$ modular p? Why is it that it's prime field $F_{p}$ forces upon $F_{p^n}$ ...
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2answers
49 views

Why are the elements of a galois/finite field represented as polynomials?

I'm new to finite fields - I have been watching various lectures and reading about them, but I'm missing a step. I can understand what a group, ring field and prime field is, no problem. But when we ...
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1answer
59 views

Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?

A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is ...
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19 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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1answer
14 views

Polynomial Residue Classes

Let $g(x) \in \mathbb{F}[x]$ be a polynomial of degree $\geq 1$. The residual class of $a(x) \in \mathbb{F}[x]$ modulo $g(x)$ is the set. $ \overline{a(x)} = \{b(x) | b(x) \equiv a(x) \mod g(x) \} ...
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37 views

Show $ \mathbb{F}_2[x]/(x^3 + x^2 + 1) \simeq \mathbb{F}_2[t]/(t^3 + t + 1)$

$x^3 + x^2 + 1$ and $x^3 + x + 1$ are both irreducible over $\mathbb{F}_2[x]$, so then we have isomorphism of fields: $$ \mathbb{F}_2[x]/(x^3 + x^2 + 1) \simeq \mathbb{F}_2[t]/(t^3 + t + 1) \simeq ...
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1answer
24 views

Polytopes in binary field

So I just stumbled across something kind of interesting. Say we're in $\{0,1\}^3$ with modulo 2 addition. The convex hull of this is the unit cube. Now, if we want to define a polytope on our cube, ...
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1answer
17 views

How can we find $[GF(p^n):GF(p)]=n$?

I was searching why $[GF(p^n):GF(p)]=n$. It is not very logical, isn't it ? I know that $$GF(p^n)=\{x\in GF(p)^{alg}\mid x^{p^n}=x \}$$ is a field with $p^n$ element since it split $X^{p^n}-X$ which ...
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1answer
62 views

How can I get a irreducible polynomial of degree 8 over $Z_2[X]$? [closed]

I have got one of degree 5: $x^5+x^2+1$, but I need one of degree 8.
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1answer
32 views

Why the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ s.t. $x\longmapsto x^p$ is surjective?

Consider the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ defined by $x\longmapsto x^p$. 1) Why is it surjective ? I recall that $\mathbb F_p^{alg}$ is an algebraic closure of ...
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2answers
34 views

A function in the integers module $p$ is polynomial. [closed]

Let $p$ a prime number and $\mathbb{F}_p$ the field of integers module $p$. Show that if $f:\mathbb{F}_p\to \mathbb{F}_p$ is a function then $f$ is polynomial.
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1answer
19 views

Relating a Character sum to a Gauss sum

Let $q$ be a prime power. Consider the mapping $f:(\mathbb{F}_q)^{\times} \to (\mathbb{F}_q)^{\times},$ where $x\mapsto x^2$. I was interested in sums of the form $$\sum_{t\in \mbox{Im}(f)} ...
1
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1answer
36 views

Isomorphic quotients of polynomial rings over finite fields

What are the elements of $\mathbb{F}_3[X]/(X^3-3)$? A similar question was posted here: Elements of the field $F_2[x] / (x^3 + x + 1)$, but it doesn't explain why the elements of that field look ...
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4answers
135 views

The equation $-1 = x^2 + y^2$ in finite fields

In an ordered field we have $x^2 \ge 0$, hence the equation $-1 = x^2 + y^2$ has no solution. But what about finite fields in general? What is the solutions set $$ -1 = x^2 + y^2 $$ of this equation? ...
4
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2answers
34 views

Splitting Field of the polynomial $x^4+x+1$ over $\mathbb{F}_2$.

What is the splitting field $\mathbb{F}_q$ of the polynomial $x^4+x+1$ over $\mathbb{F}_2$? I already knew the polynomial $x^4+x+1$ is irreducible and its roots are distinct in some extension field of ...
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2answers
49 views

Finite Fields problem [closed]

"Given a Galois Field $(\mathbb{F}, +, \cdot)$ of order 8. With an element $x \in \mathbb{F}$ we create a group $(\{x^m | m \in \mathbb{Z}\}, \cdot)$. ($x^m$ is calculated via the second operator ...
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2answers
42 views

$1+x+x^2 + \cdots + x^{p-1}$ in finite field

In $\mathbb{Q}$, $1+x+x^2+\cdots+x^{p-1}$ is always irreducible by Eisenstein criterion. What is an example of a finite field $\mathbb{F}_q$, and a prime $p$ such that $1+x+x^2+\cdots+x^{p-1}$ is ...
4
votes
3answers
52 views

Monic irreducible polynomials of degree 6 in $F_{5}[X]$

Question A How many monic irreducible polynomials of degree 6 in $F_{5}[X]$ Question B Give an example of an irreducible polynomial of degree 6 in $F_{5}[X]$ Idea for a Such a polynomial would be ...
0
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1answer
28 views

Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains 15480 elements

Question: Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains $15480$ elements Since this number is so large, I think there is some trick to get the answer. Also $15480$ is ...