Tagged Questions

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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Discrete Fourier Transform of a shift of a tuple over a finite field

Let $a = a_0 a_1 \cdots a_{N-1}$ be a sequence over a finite field $\mathbb{F}_q$, where $N \mid q^n-1$ for some $n$. Let $\xi_N$ be a primitive $N$-th root of unity in the extension ...
3
votes
4answers
139 views

How to find all irreducible polynomials in Z2 with degree 5?

I am totally lost on how to do this one. I am supposed to accomplish the following: Find all irreducible polynomials in $\mathbb{Z}_2[x]$ with degree $5$. I may use the fact that x, $x+1$ and ...
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0answers
22 views

Splitting field in finite field

What is the splitting field of the polynomial $X^{p^8}-1$ over $\mathbf F_p$? I'm confused, not is $X^{p^8}-1=(X-1)^{p^8}$ then the splitting field is $\mathbf F_p$? Thanks.
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1answer
9 views

Polynomial Factoring over a finite field

Ok, so I'm trying to figure out how to factor polynomials over a finite field. My polynomial is x^5 + x^2 + x + 1 and I have to factor over GF(2) I know the answer is (x+1)^2 * (x^3 + x + 1), because ...
0
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1answer
21 views

Find the number of primitive elements

How can i find the number of primitive elements over the field of order q? GF(27) for example. Is there a formula that I can follow? I'm really confused on how to find them. Any help would be much ...
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3answers
78 views

Proving two finite fields are isomorphic

So I'm asked to prove that $\mathbb{F}_9$, defined as $\{ a+bi$ | $a,b \in \mathbb{Z}_3,$ $i^2 = 2 \}$, is isomorphic to the field $F_1$, defined as $\mathbb{Z}_3[x]/ \langle x^2+2x+2 \rangle$, where ...
0
votes
1answer
50 views

irreducible monic polynomials

Let $P_1,P_2,\dots $ be the irreducible monic polynomials in $\mathbb{F}_p[x]$. Is there any possibility to prove the following $$\lim_{n\to \infty } \sum_{i_1+\cdots+i_n=t}z^{i_1\deg P_1 + \dots ...
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1answer
29 views

Multiplication in the Galois field GF(3^3)

I am trying to compute $x^3$ in the Galois field $\text{GF}(3^3)$ using the irreducible polynomial $f(x) = x^3 + 2x^2 + 1$. From the expression $x^3 = f(x) + (2x^2 +1)$ I proceed to take the modulus ...
4
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1answer
34 views

Suppose $F$ is a finite field of characteristic $p$. Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$.

Suppose $F$ is a finite field of characteristic $p$ ($p$ a prime). Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$. Here, $\mathbb{F}_{p}$ denotes the field with $p$ elements. Here is ...
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votes
3answers
37 views

Double finite field extension

Suppose we are given the field $\mathbb{F}_5$ and $p(X) = X^2-2 \in \mathbb{F}_5[X]$, an irreducible polynomial over $\mathbb{F}_5$. Let $\mathbb{K}$ denote the extension of $\mathbb{F}_5$ in which ...
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1answer
25 views

an irreducible polynomial over GF(2) is primitive over GF(2)

let $P \in F_{2} [X]$ of degree $7$, how to prove this: P is irreducible $\Leftrightarrow$ P is primitive i tried to use the mersenne prime !
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vote
1answer
42 views

Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
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0answers
25 views

Describe all subgroups of $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$

Is it possible to give a general discription of all subgroups of the group $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$ of inner automorphisms of $\operatorname{GL}_2(\mathbb F_{p^n})$? ...
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1answer
19 views

When Errors Go Undetected in CRC

I understand that CRC will not be able to detect errors if: The remainder of $E(x) / G(x) = 0$ $E(x) = G(x).Z(x)$ for some polynomial $Z(x)$ I understand the first point, which means that if the ...
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0answers
40 views

Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
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0answers
15 views

Why does this criterion imply that $A$ is a subfield of $E$?

$E$ is an extension field of a field $F$ and $A$ is the subset of $E$ containing all the members algebraic over $F$. "To prove that $A$ is a subfield of $E$ it is enough to show that any two elements ...
2
votes
1answer
27 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...
0
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0answers
9 views

Characterise all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$

The title pretty much explains everything. Is it possible to give an easy characterisation of all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$?
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0answers
26 views

Number of orbits of the Frobenius automorphism

Let $q=p^s$ be a prime power congruent to $1$ modulo $4$, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $\phi$ denote the Frobenius automorphism (that is $\phi(a)=a^p$ for every ...
0
votes
2answers
41 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
3
votes
1answer
39 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
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1answer
21 views

Order of an element

$\mathbb{F} = \mathbb{Z_2[x]/(x^3+x+1)}$ is a field. I need to find an element $a \in \mathbb{F}$ of order $p^n-1$ I know that $\mathbb{F}$ has order $2^3 = 8$ so $a$ must have order 7 ie, $a^7 = ...
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0answers
36 views

Solving system of equation over distinct finite fields

Is it possible to solve an equation systems which involves elements from distinct finite fields? One example could be elements from $\mathrm{GF}(2)$ and $\mathrm{GF}(2^2)$: With: $x=[1,1,0,1]$ ...
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vote
1answer
38 views

polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
2
votes
3answers
49 views

Question about the Characteristic of $\mathbb{F}_{p^n}$

We can prove that any finite field of prime characteristic $p$ must have $p^n$ elements. Conversely, let $F$ be a finite field with $p^n$ elements, where $p$ is a prime number. Is the following ...
1
vote
1answer
32 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
3
votes
2answers
45 views

How to prove a finite field is not ordered?

I have a set S={0,1}, and the addition and multiplication rules are \begin{array}{c|cc} +&0&1\\ \hline 0&0&1\\ 1&1&0 \end{array} \begin{array}{c|cc} *&0&1\\ \hline ...
4
votes
3answers
115 views

Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
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votes
4answers
32 views

Calculating $3/10$ in $\mathbb{Z}_{13}$

I'm trying to calculate $\frac{3}{10}$,working in $\mathbb{Z}_{13}$. Is this the correct approach? Let $x=\frac{3}{10} \iff 10x \equiv 3 \bmod 13 \iff 10x-3=13k \iff 10x=13k+3$ for some $k \in ...
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votes
1answer
33 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...
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1answer
45 views

Has anyone defined a limit of a sequence of fields? In particular, what is the limit of finite fields?

I'm curious about $$ \lim_{n \rightarrow \infty} \mathbb{F}_n $$ Is it $\mathbb{Z}$? That seems reasonable if you consider it as a set but of course $\mathbb{Z}$ is not a field so that is confusing. ...
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votes
1answer
45 views

Exhibiting an isomorphism between two finite fields

So I want to find the isomorphism $\phi$ that takes $F = \mathbb{Z}_3/\langle x^3 - x - 1\rangle$ to $E = \mathbb{Z}_3/\langle x^3 - x + 1\rangle$. I understand that these are both finite fields of ...
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vote
2answers
14 views

Implications of zero elemntary symmetric polynomials over a finite field

For a prime $q$ and an integer $n<q$, consider working over the finite field of $q^n$ elements. Denote by $s_n^k$ the $k$-th elementary symmetric polynomial in $n$ variables. That is, ...
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1answer
52 views

How to check field axioms given addition and multiplication tables

I need help with this question, i want to know the exact method of doing it with explanation. i am not able to get around with the logic of it.
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2answers
20 views

Homomorphisms between fields are injective.

How would I prove this? I know that I must show f(a)=f(b) => a = b I also know I must use the definition of homomorphism, ie: $f(a+b)=f(a)+f(b)$ $f(ab)=f(a)f(b)$ $f(1)=1$ I am assuming that a ...
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2answers
53 views

Automorphism of algebraic closure $\overline{{\bf F}}_p$.

Problem : I want to give an concrete example of automorphism of $\overline{{\bf F}}_p$ which fixes ${\bf F}_p$, where $$\overline{{\bf F}}_p =\bigcup_{n\geq 1} {\bf F}_{p^n} $$ and ${\bf F}_{p^n} $ is ...
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votes
0answers
118 views

On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
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votes
2answers
32 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
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0answers
32 views

$+$ and $\times $ operations in finite fields are $+$ and $\times $ $mod$ some number

I don't know how to prove this: Addition and multiplication operations in finite fields addition and multiplication $mod$ some number. Also I have another doubt that as we know that a field acts ...
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0answers
21 views

Roots of irreducible polynomials in a finite field

If $f$ and $g$ are irreducible polynomials over a finite field $\mathbb F_q$, both of degree $d$, then they both split in $\mathbb F_{q^d}$. One way to represent $\mathbb F_{q^d}$ is to adjoin a root ...
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votes
1answer
28 views

If $K$ finite field of order $p^8$ where $p\ne3$ then $\sum_{\alpha \in K}{\alpha^2} = 0$

Let $K$ be finite field of order $p^8$ where $p\ne3$ is a prime. Show that $\sum_{\alpha \in K}{\alpha^2} = 0$.
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vote
4answers
59 views

Square root for Galois fields $GF(2^m)$

Can we define a function similar to square root for $G = GF(2^m)$ (Galois field with $2^m$ elements) as $\sqrt{x} = y$ if $y^2 = y \cdot y = x$ ? For which elements $x \in G : \exists y \in G : y^2 = ...
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vote
1answer
48 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
0
votes
1answer
25 views

number of quadratic residues in finite field

Is there a way to determine how many quadratic residues are there in the finite field $F_q$ for $q = p^k$? It seems if $q=p$ exactly $(p-1)/2$ are residues and the same amount are not. Does analogy ...
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1answer
45 views

Irreducible polynomial with LI roots

I am doing a small project under the domain cryptography. Recently I stuck with a problem in mathematics related to finite fields. My question is how can I found out irreducible polynomial(or ...
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2answers
62 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
0
votes
1answer
31 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
3
votes
2answers
62 views

Jordan form of a Matrix with Ones over a Finite Field

Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$. I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and ...
1
vote
3answers
61 views

Can there be a Finite Field That Does Use Not Modular Arithmetic?

This may be a rather silly question, but I'm puzzled that (at least so far as I can tell) all finite fields use modular arithmetic. Is there no other way to construct a finite field than by defining ...
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vote
0answers
33 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...