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 $\operatorname{GF}(q)$.

learn more… | top users | synonyms (1)

1
vote
1answer
57 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: ...
0
votes
0answers
35 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 ...
-1
votes
1answer
60 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 $\zeta_{\mathbb{F}}(-1)=\sum_{...
0
votes
0answers
22 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) ...
0
votes
0answers
29 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 ...
1
vote
2answers
47 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 ...
2
votes
1answer
35 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 ...
0
votes
0answers
23 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 $γ ∈ \mathbb F_q$ ...
2
votes
1answer
481 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 $G=Gal(\mathbb{Q}(\zeta_{p})...
0
votes
1answer
37 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 ...
4
votes
1answer
34 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.
3
votes
2answers
58 views

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 $\...
1
vote
1answer
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
0
votes
1answer
28 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 ...
2
votes
2answers
94 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?
0
votes
1answer
32 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 ...
4
votes
0answers
51 views

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 ...
1
vote
1answer
26 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 why....
3
votes
2answers
89 views

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 ...
0
votes
1answer
44 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 ...
1
vote
1answer
32 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 ...
2
votes
2answers
55 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
5
votes
1answer
50 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$$a^{p+1} \in \mathbb{F}_p \subset \mathbb{F}_{p^2}?$$
0
votes
2answers
69 views

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 ...
1
vote
0answers
10 views

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?
0
votes
0answers
29 views

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 ...
3
votes
2answers
52 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) = ...
1
vote
1answer
16 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 $A$...
4
votes
0answers
36 views

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 ...
2
votes
1answer
79 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 ...
-1
votes
3answers
43 views

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 ...
2
votes
1answer
30 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.
3
votes
1answer
45 views

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 ...
0
votes
0answers
20 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?
3
votes
0answers
39 views

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, ...
0
votes
2answers
48 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 ...
0
votes
0answers
23 views

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 ...
2
votes
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 ...
1
vote
0answers
71 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
votes
1answer
42 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 ...
2
votes
1answer
33 views

Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
2
votes
2answers
49 views

Show irreducible polynomial by factorization in $\mathbb{F}_2[X]$ and $\mathbb{F}_3[X]$

I found a problem, I don't really know how to solve, although it should be something very easy, since it is stuff of Algebra I. Let $f= 29X^5−13X^4−44X^3+ 18X^2+ 35X+ 10\in\mathbb{Z}[X]$. 1) ...
1
vote
1answer
153 views

Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$

Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$. Work so far: $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$ From here, I think the Zeros of an Irreducible over a ...
1
vote
2answers
34 views

Discrete math: find prime number which solves the following conditions

Find prime number $p$ and polynomial $f$, so that the ring $F_p[x]/(f)$: 1) contains non-zero element $w$: $w^n=0$ for some $n$ 2) doesn't satisfy 1), but is a field 3) is a field which contains $8$...
0
votes
1answer
55 views

How to find the elements of a finite field?

OK, I am asked to find the elements for the addition and multiplication tables for the finite fields with eight elements. I know I've asked about this question previously, and I've almost gotten a ...
0
votes
0answers
36 views

Cyclic code and its generating polynomial

Can someone give me an example of a $[3,2]$ linear cyclic code and its generating polynomial? Does this mean it is a binary cyclic code with length 3? Also how do I find all binary/ternary cyclic ...
1
vote
0answers
21 views

Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface $$X=\{...
1
vote
1answer
48 views

Proving the Existence of Finite Fields By Counting the Number of Irreducibles

I want to prove that given a prime $p$ and a positive integer $n$, there is a finite field of order $p^n$. I want to do it by showing that there is an irreducible polynomial of degree $n$ over $\...
0
votes
0answers
27 views

Question about subfield subcode of a cyclic code

If $C$ is a cyclic code over $\mathbb{F}_{q^n}$, then $C|_{\mathbb{F}_q}$ is a cyclic code over $\mathbb{F}_q$. I know this holds for a linear code, what about a cyclic code? Thanks in advance
1
vote
0answers
29 views

What are the cardinalities of the similarity classes of matrices over a finite field?

Similarity of matrices forms an equivalence relation. The set of linear maps between two finite dimensional vector spaces over a finite field is finite. Is there a combinatorial formula for the ...