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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existence of irreducible polynomial of degree 10 over finite field

Prove that there exists an irreducible polynomial of degree 10 over the the field of 25 elements. I know that the multiplicative group of non-zero elements of any finite field is cyclic. So how can I ...
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51 views

How do I find the ideals in the ring $\mathbb F_3[x]/(x^2+2)$?

Clearly $\{0\}$ and $\mathbb F_3[x]/(x^2+2)$ will be ideals. How would I find the others?
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53 views

Any method to solve this system of equation?

We have m variables $ x_{1},x_{2},...,x_{m} $ which are elements of field $F_{p}$ and we are given m equations of the form $$\sum_{i=1}^{m} x_{i}^{n} = c_{n} \mod p \qquad for \: 1 \le n \le m$$ ...
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1answer
40 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
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21 views

I am currently working on the implementation of the Powerline System which is based on the Chor-Ri

I am currently working on the implementation of the 'Powerline System' which is based on the Chor-Rivest cryptosystem for my number theory project. There is a step in the key-generation phase of the ...
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1answer
33 views

Proving that if the cardinality of a field $F$ is finite and equal to $q$, then the ring $F[X]/(X^n)$ is finite of cardinality $q^n$

I'm trying to prove how the cardinality of a field $F$ is finite and equal to $q$, then the quotient ring $F[X]/(X^n)$ is finite of cardinality $q^n$. How do I go about this when the quotient ...
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7 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
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62 views

Does NTT have an upper bound?

I'm working with NTT (Number theoretic transform) to reduce the complexity of a polynomial multiplication. For this, I'm using $P = 2^{64}-2^{32}+1$ to generate the primitive root $\omega_N$ needed by ...
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2answers
61 views

Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
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44 views

Totien-Sum: why GCD( {n}/d, q/d) = 1; implies Sum{Totient(d/q) } = q

Have seen answer to this question. still don't understand.. Totient sum is defined: q = Sum(Totient (d) ); sum on all d : d|q More specific; The proof has these steps: 1. If d is a divider ...
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1answer
44 views

I don't understand a step of the proof that a nonzero finite commutative ring with no zero divisors is a field.

This seems to be a fairly classic problem in algebra: Let $R$ be a nonzero finite commutative ring with no zero divisors. Prove that $R$ is a field. Here is the a solution I came across: It has ...
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29 views

General Multiplication Algorithm for Finite Field of GF(p^m) for small values of p^m

I am working on a small computer program to generate multiplication table of general finite field in form of GF(p^m) where p is a prime number and m is an integer greater than one. While the special ...
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1answer
59 views

Are zeta functions discussed over finite fields? [closed]

Let $\mathbb{F}$ be a finite field. I wonder if someone discussed the behaviour of the analogous of zeta functions over $\mathbb{F}$? For example, one can easily see that ...
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21 views

Uniformly Random Polynomial

Hypothesis: all the polynomials and values are defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number. My question is related to information security. Assume (in a protocol) ...
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27 views

Number of solutions to $x^2-y^2=a$ over a finite field.

Let $F=\mathbb{F}_{q}$ be a finite field where $q$ is an odd prime power. Fix $a\in F\setminus\{0\}$. I would like to find out the number of solutions to the equation $$x^2-y^2=a.$$ Could anyone give ...
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1answer
24 views

Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field.

Given Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field of $3$ elements then, $F$ is a field with $27$ elements. $F$ is separable but not a normal extension of $F_3$. The ...
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1answer
32 views

Balanced incomplete Block design for testing an experiment

I am reading something balanced incomplete block design from a book. I don't understand why is it easy to see that in this design Each vehicle is evaluated 8 times, each test driver evaluates 4 ...
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22 views

Let $α$ be an element of $\mathbb F_q$ with largest order $t$ . Show that set of elements with order dividing $t$ should have cardinality $t$

Let $α$ be an element of $\mathbb F_q$ with largest order $t$ . Show that set of elements with order dividing $t$ should have cardinality $t$, so there must be some non-zero element $γ ∈ \mathbb F_q$ ...
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1answer
261 views

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then ...
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1answer
36 views

Find all the elements that are fixed by $Frob_3$

Consider the finite field $\mathbb F_9 = \mathbb F_3[x]/ \langle x^2 + 1\rangle$, and recall the Frobenius isomorphism $Frob_3 : \mathbb F_3 → \mathbb F_3$, given by $Frob_3(x) = x^3$ Find all the ...
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32 views

Is $t^4 + 7$ irreducible over $\mathbb{Z}_{17}$?

There aren't any linear factors, $-7$ doesn't have a square root in $\mathbb{Z}_{17}$ so I've ruled out those 'easy' factorisations. I'm not sure how to continue. Looking for a hint, not a solution.
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Any two Singer cyclic subgroups of GL(n,q) are conjugate

Cyclic subgroups of $\operatorname{GL}(n,q)$ of order $q^n - 1$ are called Singer cyclic subgroups. The following statement seems to be well-known: Any two Singer cyclic subgroups of ...
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21 views

Is a finite field matrix treated the same as normal matrix?

Do I bring a finite field matrix to RREF the same way as a real number matrix, or do I have to follow the finite field addition and multiplication operations
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1answer
27 views

how many different outcomes for the inner product over finite fields

defining the inner product $<u ,v >=\sum_{i=1}^ku_iv_i (mod \ p)$ when $u,v \in GF(p)^k$ ($p$ is a prime). for any non-zero, fixed $u$ and for every $v$ how can I show that the outcome will be ...
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89 views

Geometric reasons finite fields have prime power orders?

All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
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29 views

$L \supset K$, $K$ has characteristic $3$, $[L:K]=3$. Find an example where $L$ is separable/non-separable.

Question: $L \supset K$, $K$ has characteristic $3$, $[L:K]=3$. Find an example where $L$ is (a) separable (b) non-separable. What I know: $L$ is a finite field extension of $K$. So, $K$ is its ...
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Moments of the number of roots of polynomials over finite fields

Let $F:=\{f\in\mathbb{F}_q[X_1,\ldots,X_n]: \textrm{deg}(f)\leq d\}$ be the set containing all $n$-variate polynomials of degree less than or equal to $d$ over finite field $\mathbb{F}_q$ of prime ...
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20 views

Expected number of rows of the full rank matrix

Let A be a m by n random matrix over finite fields F_q. Suppose the rank of A is n. How much does expected number of m? I think m maybe qlogq by bins and balls property But I do not know exactly ...
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Let $\mathbb{F}_2 \cong \mathbb{Z}/2\mathbb{Z}$. Is $x^4+x^2+1$ irreducible in $\mathbb{F}_2[x]$?

Let $\mathbb{F}_2 \cong \mathbb{Z}/2\mathbb{Z}$ denote the field of 2 elements (a) Is $x^4+x^2+1$ irreducible in $\mathbb{F}_2[x]$? Find a complete factorization. (b) How many irreducible ...
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38 views

Solve the equation $x^n=a$ in a finite field

Good evening! I'm studing something about finite fields and I didn't find anything that can answer my question. I have a finite field $\mathbf{F}_q$ and the equation to solve is $x^n=a$. Which are ...
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1answer
29 views

Non-zero coefficients of primitive polynomials

Let $R$ be a finite field of $q$ elements, $m,n\in \mathbb{N}, 2\leq m, 2\leq n$. I want to prove that there exists a primitive polynomial $$F(x) = x^{mn}-\sum\limits_{j=0}^{mn-1}f_jx^j\in R[x]$$ with ...
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42 views

The number of subspace in a finite field

How to prove this conclusion If V is a vector space of dimension n and F is a finite field with q elements then number of subspace of dim k is
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46 views

Element of $\mathbb{F}_{p^2}$ of order $p^2$ raised to $p+1$th power is element of $\mathbb{F}_p$?

For any prime number $p$ and any element $a$ of the finite field $\mathbb{F}_{p^2}$ of order $p^2$, do we have$$a^{p+1} \in \mathbb{F}_p \subset \mathbb{F}_{p^2}?$$
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Let F be a finite field of characteristic $p$. Show $f(a) = a^p$ is a ring homomorphism, injective, and surjective

Let F be a finite field of characteristic $p$. Show that the function $f:F \to F$ defined by $f(a) = a^p$ is a) a ring homomorphism, b) injective and, c) surjective. I tried to approach this problem ...
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Finding a prime in range with the largest minimal non quadratic residue

Given k, I'm trying to find the k-bit prime that has the largest minimal non-quadratic residue. I was wondering if there's any construction like that. Perhaps some use of the CRT?
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Notation used for equations in finite fields

In some work I am currently writing there is a point at which I have to put values into a few different formulae. I am operating over a finite field so I aim to use values in this field. As an ...
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41 views

How do I find the order of an element if I am given the minimal polynomial of the element?

For example, let's say I am given an element $α$ in a field of characteristic $2$. Further, I am given the minimal polynomial of α with respect to $GF(2)$. Let's say that minimal polynomial is $f(x) = ...
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1answer
14 views

Finding the solution of smallest magnitude involving a non-injective matrix

Let $A$ be an $n \times n$ matrix that is non-injective, specifically one where the entries are in $\mathbb{Z}_2$ or, equivalently, $GF(2)$. Let $b$ be an $n\times 1$ matrix that is in the image of ...
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Finite Fields and Discrete Log Problem

The finite field $\mathbb F_{p^p}$, for a prime $p$, can be described as $\mathbb F_p[x]/(x^p−x−1)$. Let $G=\mathbb F_{p^p}^\times/\mathbb F_{p}^\times$. Estimate the cost of obtaining the discrete ...
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78 views

Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
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What are the elements of GF(9)? What is the 'addition' and 'multiplication' operations on this field?

I have read methods to construct GF(p^m). I have understood the primitive polynomials and other concepts but I have not understood how the p and m are entering into the discussion. And finally the ...
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1answer
28 views

Is a finite field algebraic extension of a fixed degree unique or just unique up to isomorphic?

well,if $[F_{q}(\alpha):F_{q}]=m$, can i prove that $F_{q}(\alpha)=F_{q^{m}}$ ? or i can only prove that $F_{q}(\alpha)$ is isomorphic to $F_{q^{m}}$ ? thanks for advance.
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Action on its generators of splitting field of $x^4 +5$

Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$ Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$. $x^4 +5$ is irreducible so there is one orbit ...
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17 views

I have vectors over a finite field GF(p) where p is power of a prime. How are all the different ways in which I can define the length of a vector?

I have vectors over a finite field GF(p) where p is power of a prime. What are all the different ways in which I can define the length of a vector?
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a symmetric matrix over GF(2)

let $A$ be a symmetric $n$ by $n$ matrix over $\mathbb{GF}(2)$. Using elementary linear algebra, it is quite easy to show that diag $A$ is in the range of $A$, where diag $A = [a_{11},a_{22}, \dots, ...
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46 views

Polynomials over a field having equal polynomial functions

Let $p(x),q(x)$ be two polynomials over a field $F$ such that $p(a)=q(a)$ for all $a\in F$. Can we say that always $p(x)=q(x)$? If $F=\mathbb Z_5$, then it is possible to find examples of $p(x)\neq ...
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Want to know the best books on Matrix algebra dealing with the special case of over GF(2).

Want to know the best books on Matrix algebra dealing over GF(2). Want to know the special properties of matrices over GF(2). For example, in GF(2), non-zero vectors can have a dotproduct with ...
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2answers
34 views

Proving that $x^4+x^3+1$ has no solution over $\mathrm{GF}(2^e)$, $e$ odd.

I'm trying to find out whether $x^4+x^3+1$ has a solution over $\mathrm{GF}(2^e)$, $e$ odd. Some quick calculations for small values of $e$ indicate that it does not, but my background is not in ...
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49 views

Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
2
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1answer
36 views

Easy way to find the order of elements in a finite field

I am trying to work out the multiplicative order of each non zero element in $F_7$. Lets say I am looking at the number $3$. I know its order is $6$. Instead of having to work out the powers of ...