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)

0
votes
0answers
48 views

Irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$ [on hold]

We wish to find all the irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge ...
0
votes
0answers
30 views

Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$

Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$ I am trying to solve this question which looks like to be ...
1
vote
0answers
18 views

Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
0
votes
1answer
23 views

Finding $|E|$, where $E$ is the Splitting Field of $x^8-1$ over a Field of $4$ Elements.

This is my attempt to find $\vert E \vert$, which is the order of the field $E$. If I am on the wrong track, please guide me to a technique that will work with more general fields and polynomials. We ...
2
votes
2answers
43 views

Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
5
votes
2answers
108 views

When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?
1
vote
1answer
83 views

Hitting a line in a $d$ dimensional cube

Let $F$ be a finite field of order $n$, and let $d$ be an integer. A line in $F^d$ is a function $\ell: F \to F^d$ given by $\ell(t) = x + t*h$, where $x,h \in F^d$, $h \neq 0$, and $t*h = (tx_1, ...
2
votes
1answer
47 views

Properties of the finite field with $729$ elements

I am trying to solve the following problem: let $K$ be a finite field with $729$ elements. How many $\alpha\in K$ make $K^* = \langle \alpha\rangle$? How many fields $E$ are such that $K|E$ is a ...
4
votes
1answer
33 views

Find a $u$ so that $k(u)$ is the fixed field of $φ$, determine the minimal polynomial over $k(u)$

Let $k = F_p$, and let $k(x)$ be the rational function field in one variable over $k$. Define $φ : k(x) \to k(x)$ by $φ(x) = x+1$. Show that $φ$ has finite order in $Gal(k(x)/k)$. Determine this ...
4
votes
3answers
110 views

Iterated square roots over finite field. When do we hit a nonresidue?

Suppose that we are working within the integers modulo $p$ where $p$ is some odd prime number. Suppose that $x_0$ is a (nonzero) quadratic residue mod $p$ then there exists some $x_1$ such that $x_1^2 ...
2
votes
2answers
46 views

homogeneous polynomials over finite fields

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?
-3
votes
2answers
46 views

Exercise in arithmetic of a finite field

I am in difficult in resolving this exercise in Galois Theory : "in $GF(2^5)$ calculates the product $(1,1,1,0,1)(0,1,0,1,0)$ , generator of $GF(2^5)^*$ ". I don't know how to proceed.. thank you
0
votes
2answers
71 views

The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
3
votes
3answers
79 views

How many elements does this ring have?

I know that the following ring is not a field because the defining polynomial is reducible into two polynomials that are irreducible: $$\mathbb Z_2[X]/(x^5+x+1)$$ where $$x^5 + x + 1 = (x^2 + x + 1) ...
-2
votes
1answer
32 views

Number of subfields of a given field [duplicate]

Let $F$ be a field with $5^{12}$ elements. What is the total number of proper subfields of $F$? A) $3$ B) $6$ C) $8$ D) $5$ Explain the concept used to solve the question.
1
vote
0answers
31 views

Difference between $f(.)$ over $\mathbb{Z}_q$ and $f(x)$ for $x \in\mathbb{Z}_q$

What is the difference between the two following notations. function $f(.)$ over $\mathbb{Z}_q$ . function $f(x)$ where $x$ takes values from $\mathbb{Z}_q$ I think that both are same. Is there ...
4
votes
1answer
64 views

Summing up the elements of a finite field. [duplicate]

I want to show that $\sum_{x\in \mathbb F_{q}}x^i=0$ if $q-1$ does not divide $i$. Can someone give me a hint?
0
votes
0answers
27 views

Square root in a general field

In $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ there are obvious ways to calculate the square root of a quadratic residue. For finite fields of order $p$ we can use the Tonelli–Shanks algorithm. How ...
4
votes
1answer
38 views

For which $n \in \mathbb{N}$ $f(x) = x^{2n}+x^n+1$ is irreducible in $\mathbb{F}_2[x]$?

I have $$f(x) = x^{2n}+x^n+1 \in \mathbb{F}_2[x].$$ When is this polynomial irreducible? It is obvious that for even $n$ this polynomial is reducible. But I don't have any idea about odd $n$.
2
votes
2answers
43 views

Are there any books/papers talking about inner product on vectors over finite fields?

Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product ...
2
votes
2answers
27 views

$\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$

I want to find $\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$ So then this will depend on the field, but no harm in direct computation for arbitrary matrices: ...
1
vote
1answer
30 views

Derivative of polynomial in GF(16)

how can I find the derivative of the following polynomial in $GF(2^4)$: $\alpha x^4+x^3+\alpha x^2+\alpha^2 x+1$ ?
0
votes
1answer
13 views

Can the null space of a sparse matrix have a set of basis that is sparse?

Suppose we have a linear system $Ax=b$ in GF(2), where $A$ is sparse. Its solution is $x = Tz + x_0$. It is apparent that $T$ is not unique, then is there a way to find a T that is as sparse as ...
1
vote
0answers
45 views

Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
3
votes
1answer
72 views

Does the equation $x^3-y^3=b$ have solutions in a finite field $F$ where $b\in F$?

Does the equation $x^3-y^3=b$ has solutions in a finite field $F$ where $b\in F$? I know it is true for the equation $x^2-y^2=b$, as for the case $\text{char}\ F \neq 2$, $x^2-y^2=(x+y)(x-y)=b.1$, ...
1
vote
1answer
28 views

A question regarding proving the fact that every finite field is perfect

I am trying to prove the fact that every finite field is perfect. Hence, every irreducible polynomial is separable (does not have a repeated root). This is easy to prove when in a field of ...
1
vote
0answers
29 views

Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = ...
3
votes
0answers
64 views

Working in a field.

When I calculate vector spaces, diagonalization of matrices, linear transformations on a field, can I work in $\mathbb R$ and ultimately transform the result to that field? For example if I calculate ...
3
votes
2answers
38 views

Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings?

Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings? If so, write down an explicit isomorphism. If not, prove they are not. My Try: Since $x^2+2$ is ...
0
votes
2answers
21 views

Best way to calculate unit roots of GF(n)

What's the best and most simple way to calculate unit roots of $GF(n)$. $n$ could be any integer. Please make a distinction between primes and non-primes. Example: Show that $GF(29)$ has 7th ...
-1
votes
1answer
38 views
2
votes
0answers
20 views

Show that polynomial is primitive in GF(5) [duplicate]

How can I show that $x^2 + 2x + 3$ is primitive in $GF(5)$? My idea: $ x^1 = x\\ x^2 = -2x - 3 = 3x + 2\\ x^3 = (3x + 2)x = 3x^2 + 2x = 3(3x + 2) + 2x = x + 1\\ ...\\ x^a = 1\\ $ This would take ...
2
votes
0answers
20 views

Distribution of Primitive Elements Finite Fields Prime Order

It is well known that the integers modulo a prime $p$ form a finite field and that the multiplicative group of this finite field is cyclic, with $\phi(p-1)$ different possible choices of primitive ...
2
votes
1answer
35 views

Finite fields - quadratic extensions

Let $F_q$ be a finite field of $\operatorname{char}F\neq2$. suppose $a_1$,$a_2$,...,$a_q$ are the elements in $F_q$. show there exists $i$ such that for any $j$ $a_i\neq a_j^2$. compute the degree ...
0
votes
2answers
44 views

When decoding a block code, how do you know which error a syndrome corresponds to?

I'm working with forward error correcting block codes such as Hamming(7,4) and Golay(23,12). I'm quite new to this field, so there are some things that I don't yet understand. I chose these codes ...
1
vote
1answer
33 views

We can code the integers into the orders at zero of elements of $F_q(t)$

There is the following part in the paper that I am reading: We can code the integers into the orders at zero of elements of $F_q(t)$ (the field of rational functions in $t$ with coefficients in a ...
2
votes
1answer
35 views

Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.

Consider the map $\varphi:\mathbb{F}_{p^{n}} \rightarrow \mathbb{F}_{p^{n}}$ defined by $x \mapsto x^{p}-x$. Since $(a+b)^{p}= a^{p}+b^{p}$ for all $a,b \in \mathbb{F}_{p^{n}}$ we have that $\varphi$ ...
0
votes
2answers
34 views

additive character of a finite field, trace map to middle field

The additive character of a finite field $\mathbb F_q$ is obtained by using the trace function to base field $F_p$. Can we write some of them into a middle field. Using trace function from $\mathbb ...
1
vote
2answers
72 views

The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
2
votes
3answers
145 views

Permutations of the elements of $\mathbb Z_p$

Let $p$ be prime. Describe all permutations $\sigma$ of the elements of $\mathbb Z_p$, having the property that $\{\sigma(i)-i: i\in\mathbb Z_p\}=\mathbb Z_p$ (Added by Robert Lewis in an attempt ...
3
votes
2answers
41 views

Maximal unramified extension of a global function field

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?
1
vote
3answers
53 views

What is the definition of a $\mathbb{F}_2$-linear function?

To clarify, the function is $f:\mathbb{F}_2^m\rightarrow \mathbb{F}_2$. So, does it mean linear in each variable, or perhaps that each monomial is of degree $\leq1$? I know that sometimes terms have ...
1
vote
1answer
26 views

What can be said about a matrix with a constant column of ones with entries from a finite field?

I am working with matrices of the following structure: $A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ ...
4
votes
1answer
83 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
1
vote
1answer
45 views

For which elements $t$ in a finite field $\mathbb{F}_{p^n}$ is $t^2 - 4$ a square?

That is, how to characterize the elements $t \in \mathbb{F}_{p^n}$ for which there exists $x \in \mathbb{F}_{p^n}$ such that $t^2 - 4 = x^2$?
2
votes
1answer
46 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
1
vote
1answer
37 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
0
votes
2answers
31 views

Power in finite field

Does the following statement hold true for any finite field? $$a^p\equiv a \qquad(\mathbb{Z_p})$$ I have tought at it this way: all numbers in $\mathbb{Z_p}$ are $\in \{0,\mathbb{Z_p}\}$ and ...
2
votes
1answer
16 views

Incident vector for lines in a 2D-Euclidean Geometry over Finite field

Consider the 2-D $EG(2,2^2)$ geometry. Let $\alpha$ be a primitive element of $GF(2^{2\times 2})$. The incident vector for the line $\mathcal{L} = \{\alpha^7, \alpha^8, \alpha^{10}, \alpha^{14}\}$ is ...
1
vote
0answers
34 views

$x^p -x-1$ irreducible over $\mathbb{F}_{p}$ [duplicate]

Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$. I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is ...