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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Field construction
Explain how to construct a field of order $343$ not using addition and multiplication tables.
I understand that every finite field has order $p^n$ for some prime $p$. Since $343$ is $7^3$, let ...
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2answers
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System of equation over GF(211) (corrected)
I have this system of equation.
$a+b+c=171, a+2b+4c = 46, a+3b+9c = 170$.
My task is to solve this system over $GF(211)$.
Is there any special process?
Thanks for advice.
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1answer
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Irreducible polynomials over $F_q$ with exponents of the form $q^k - 1$.
Let $q$ be some prime power. Is there an explicit family of irreducible polynomials in $F_q[X]$ of the form $\sum_j a_j X^{q^j - 1}$? Thanks!
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Problem related polynomial ring over finite field of intergers
if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
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51 views
Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.
Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points.
Using Fermat's ...
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Find the inverse of $\alpha^{38}$ in $\mathbb F = \mathbb Z_2[x]/\left<x^4+x+1\right>$
Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$
$\alpha^4=\alpha+1$
$\alpha^5=\alpha^2+\alpha$
... (the rest ...
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Encode the message $[1,1,0,1,1,0,1]$ in BCH code based on the field $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1}$
So here's what I understand so far:
$\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1} = GF(16)$
The code is written as $[x^{14},x^{13},x^{12},x^{11},x^{10},x^{9},x^{8}$ $|$ ...
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52 views
Express each power of the root $\alpha$ of $\frac{\mathbb Z_2[x]}{\langle x^3+x^2+1\rangle}$ as linear combinations of $1, \alpha$ and $\alpha^2$
There are $8$ elements in $\frac{\mathbb Z_2[x]}{\langle x^3+x^2+1\rangle} = GF(8)$
and this generates the set $\{0,1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$
We're required to express $\alpha^1$ all the ...
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109 views
A Finite Field of Order $125$ has a Subfield of order $25$?
How to prove that every finite field of order $125$ has a subfield of order $25$. In general what is the strategy to attack such kind of problems?
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Finite Fields of Order 3
I have been trying to learn Real Analysis, though I'm having trouble with a problem.
Show that there exists one and (essentially) only one field with three elements.
Any help will be appreciated.
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1answer
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factorization of the cyclotomic $\Phi_n(x)$ over $\Bbb F_p$
One can prove that the $\Phi_n(x)$ are irreducible over $\Bbb Z$. Where $\Phi_n(x)=\prod _{(a,n)=1}\zeta_n^a$ (i.e the product of the primitive n-rooth of unity). I want to find a factorization of ...
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Showing that the oder of the group and all the element is the same.
If $f=x^n-1 \in K[x]$, $L$ splitting field of $f$ and $\gamma\in L$ the generator of $H= \{\alpha \in L \mid \alpha^n=1\} , m=|H|$,
how can I show that $m|n$, should it depend on the characteristic ...
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Ideals of the ring $\mathbb{F}_q[X]/(X^n-1)$
I need little help in proving the following result :
Consider the ring $R:=\mathbb{F}_q[X]/(X^n-1)$, where $\mathbb{F}_q$ is a finite field of cardinality $q$ and $n\in\mathbb{N}$. Then any ideal $I$ ...
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153 views
LFSR (Linear Feedback Shift Register)
Given polynomial $P(x)=x^6+x^3+1$ belonging to $\mathbb{Z}_2[x]$. Build an $LFSR$ corresponding to $P(x)$. Then find the maximal period of its output sequence and the initial state that could lead to ...
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2answers
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Showing that Trace($ax$) = $0$ for all $x$ in field $F$ of order $2^N$ implies $a = 0$.
How can I show that Trace $(ax) = 0$ implies that $a = 0$ in a field $F$ (over F2) of order $2^N$? I get that I can something like the following:
Trace($ax_1)=$ Trace($ax_2$) $\implies$ ...
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1answer
104 views
A question on the irreducible divisors and splitting field of $x^{p^n} - x\in \mathbb F_p[x]$.
I need to prove that any irreducible polynomial $f$ of degree $d\,\big|\,n$ over $\mathbb F_p$ devides $x^{p^n} - x$. I know that the splitting field of the latter is the finite field with $p^n$ ...
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513 views
How do i calculate a multiplication table for GF(8)?
Could you please provide the steps involved in calculating a multiplication table for GF(8)?
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3answers
304 views
Characterization of irreducible polynomials over finite fields
How much is known about irreducible polynomials over finite fields? I have seen the formula (a result of Mobius inversion) that gives the number of such polynomials, but I am looking for something ...
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1answer
142 views
Using Lagrange's Interpolation Formula to show that boolean functions over finite fields are polynomials
Let $F_2$ be the set of all the functions from the finite field $GF(2^n)$ of $2^n$ elements to $GF(2)$. I am reading a textbook that proves that the elements of $F_2$ can be represented by ...
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51 views
The probability of sum of $k$ randomly chosen numbers from $\mathbb{Z}_p$ being zero
Let $\{x_i\}_{i \in [k]}$ be randomly chosen independently from the uniform distribution over $\mathbb{Z}_p$.
What is the probability that $\Sigma_{i\in [k]} x_i = 0$?
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1answer
85 views
The number of primitve elements in $GF(s^2)$
The multiplicative group of any finite field is cyclic, a generator of this group is called a primitive element of the finite field. Is it true that the number of primitive elements in the finite ...
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root of $x^3+x+1$ over $\mathbb{F}_5$
What is the simplest radical expression of some root $a \in \overline{\mathbb{F}_5}$ of the polynomial $x^3+x+1 \in \mathbb{F}_5[x]$? I wonder if one can simplify the general formulas in this special ...
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183 views
inverse,multiplicative inverse and Congruence of a prime field
I am dealing with ECC in these days which heavily based on finite fields. I want to how to find a inverse of a value in finite field and what is multiplicative inverse and also Congruence
F29-
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113 views
finite fields, a cubic extension on finite fields.
Let $q=p^n$ where $p$ is a prime number. For which $q$ is the extensión $\mathbb{F}_{q^3}/\mathbb{F}_q$ (i.e the extension of degree $3$) an extension of the form $\mathbb{F}_q(\root 3 \of \alpha )$
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3answers
84 views
Possible Cardinality of a Field
The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$.
...
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$Y^3$ congruent to $1 \pmod {p}$
How to get the condition on $p$ for which $y^3$ congruent to $1$ modulo $p$ has $3$ solutions ( $1$ solution $x= 1$ is always possible, right ?).
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Automorphisms of $\mathbb{F}_2[x,y]$
What are the automorphisms of the 2-variable polynomial ring over $\mathbb{F}_2$, the field with 2 elements? Are they generated by $(x \mapsto y, y \mapsto x)$, $(x\mapsto x+ p(y), y\mapsto y)$, and ...
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Cryptography: how to determine if an element is in a finite field?
I'm working on a cryptography project that is basically a semantically secure modification to ElGamal. My question is: how do I determine if an element is in a finite field without calculating the ...
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Irreduciblity of the polynomial $x^{p^n}-x+1$ [duplicate]
Possible Duplicate:
Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
What are the values of $n$ for which the polynomial $$f(x):=x^{p^n}-x+1$$is irreducible ...
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Alternative solution to determine the number of irreducible, monic polynomials in $\mathbb{Z}_p[x]$ of degree $k$
I know the problem of the number monic, irreducible polynomials of degree $k$ in $\mathbb{F}_p$ have been discussed and that there is a general formula which solves this problem. Nevertheless, I have ...
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0answers
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when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?
If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
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1answer
170 views
Proving a polynomial irreducible over finite field [duplicate]
Possible Duplicate:
How do I prove that $x^p-x+a$ is irreducible in a field of $p$ elements when $a\neq 0$?
How to prove that $x^p - x - 1$ is irreducible over $F_{p}[x]$.
I thought ...
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2answers
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Finding a polynomial $g$ such that $g^2=f$ for certain $f$ in $\mathbb{F}_{16}$
Let $L = \mathbb{Z}_2[x]/\langle x^4 + x + 1 \rangle$ and $\alpha := [x] \in L$.
I want to find $g \in L[y]$ with $g^2 = f$ and
$$f = (\alpha^2 + \alpha)y^8 + \alpha y^4 + \alpha^3y^2 + \alpha + 1 ...
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1answer
56 views
Efficient method to determine if a set of vectors span a finite field with some constraints on the constants.
In a finite field $\{0,1,2\}^2$, given a set of vectors $[0\:1],[1\:0],[1\:1],[2\:2]$, we can have the linear combination, $c_1[1\:0]+c_2[0\:1]+c_3[1\:1]+c_4[2\:2] = [s_1\:s_2]\in\{0,1,2\}^2$, where ...
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1answer
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conjugacy classes in representation theory
I have a question on conjugacy classes in this post, especially to this sentence:
"if $g$ is a rotation of order $5$, then it is $K$-conjugate to each of $g^{13},g^{13^2},g^{13^3},g^{13^4}=g$".
...
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1answer
147 views
program to find matrix group given generators (Matlab)?
Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group?
The best I've ...
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301 views
hash function for matrices over finite field (Matlab)?
What is a good hash function for small nonsingular matrices over a field $\mathbb{F}_p$ for $p$ prime? I'm looking for an integer function which is close to being injective (but not necessarily ...
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2answers
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Irreducible polynomial over field of order p
Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$.
I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.
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1answer
277 views
Reed-Solomon Code calculation
I have a Reed-Solomon Code which can correct t=2 errors. The generator polynomial is $p(X) = X^3 + X + 1$ and $p(a) = a^3 + a + 1 = 0$ this means $a^3 = a + 1$
What is the degree of generator ...
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6answers
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How to construct minimal polynomial?
This is an exam question from last semester.
We have the finite field
$$ \mathbb F_{81} = \mathbb Z_3 [x]/(x^4+x^2+x+1)$$
(a) Prove that the polynomial $$ x^4+x^2+x+1 $$
is irreducible
(b) ...
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2answers
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How to determine if this is a field?
A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$).
The question is:
True or False: The ring $R$ must be a field.
I thought that if $R$ was a field it had to be a finite ...
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$\mathbb{Z}/n\mathbb{Z}[T]$ and zero divisors
I read the following in wiki, but I can't understand what is meant by "divisor" there.
Notice that $(\mathbb{Z}/2\mathbb{Z})[T]/(T^{2}+1)$ is not a field since it admits a zero divisor ...
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1answer
263 views
Do there exist vector spaces over a finite field that have a dot product?
I've stumbled in the german wikipedia over the question if a vector space over a finite field exists, that has a dot product.
Definition of dot product
A dot product over a $\mathbb{K}$-vector space ...
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Can you construct a field with 6 elements? [duplicate]
Possible Duplicate:
Is there anything like GF(6)?
Could someone tell me if you can build a field with 6 elements.
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4answers
230 views
Number of solutions of $x^2_1+\dots+x^2_n=0,$ $x_i\in \Bbb{F}_q.$
Let $\mathbb F$ be field with $q$ element and $\operatorname{char}(\mathbb F)\neq2$. I want to know about the number of solutions of this equation:
$$x^2_1+x^2_2+\dots+x^2_n=0 \text{ where } x_i\in ...
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5answers
88 views
$(x-1)^{p-1} = 1+x+\dots+x^{p−1}$ mod p?
It is well known that in characteristic p we have the "freshman dream" $x^p -1 = (x - 1)^p$. Some experimentation seems to suggest that the heuristic computation $(x-1)(x-1)^{p-1} = (x-1)^p = x^p - 1 ...
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1answer
80 views
If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field
Let $R$ be a commutative unitary ring and suppose that the abelian
group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal
of $R$.
Then $R/P$ is a finite field.
Well, the ...
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4answers
103 views
nonzero elements of splitting field
Let $F$ be a splitting field of $x^{p^{n}} - x \in \mathbb{Z}_p[x]$.
How is it that the nonzero elements multiply to $-1$ and sum to $0$? I don't get how we get that result.
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1answer
63 views
Latin squares of even order with all cells only participating in one subsquare.
For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below:
...
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3answers
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Uniqueness of prime-power fields
I'm still stuck on the proof of the following theorem. I've asked two questions so far to get to where I am even at this point.
Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and ...
