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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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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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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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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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 Möbius inversion) that gives the number of such polynomials, but I am looking for something ...
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1answer
370 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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58 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
135 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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1answer
646 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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1answer
195 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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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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1answer
97 views

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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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
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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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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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203 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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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
315 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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1answer
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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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190 views

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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759 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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Is there a finite field in which the additive group is not cyclic?

Is there a finite field whose additive group is not cyclic?
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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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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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102 views

$\mathbb{Z}/n\mathbb{Z}[T]$ 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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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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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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$(x-1)^{p-1} = 1+x+\dots+x^{p−1}$ 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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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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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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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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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 ...
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Can't follow a proof involving Prime-Power Fields

Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and $E'$ are fields of order $p^{n}$, then $E\cong E'$. Proof: Both $E$ And $E'$ have $\mathbb{Z}_{p}$ as prime fields (up to ...
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Is the Algebraic Closure of a Finite Field Algebraically Closed?

A Lemma stated: Let $F$ be a finite field, of prime characteristic $p$, and with algebraic closure $\overline{F}$. The polynomial $x^{p^{n}} - x$ has $p^{n}$ distinct zeros in $\overline{F}$. The ...
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Problem Involving Finite Fields

I've arrived at a Theorem in text that I'm confused about: Note: My question below is about the statement of this theorem, not about a proof for it. (The proof is supplied in the text) Theorem: ...
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Question about a corollary about Finite Fields

Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$. Theorem: Let $E$ be a degree $n$ finite ...
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Order of an element in $\mathbb{F}_q$ where $p^k = q$

I apologize if the title seems to be misleading as I couldn't conjure up a more relevant title. My question is that suppose we have that a prime $p$ and $q = p^k$ for some positive integer $k > 1$. ...
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Irreducible Polynomials in Finite Fields

I'm reading through some notes online concerning finite fields, and attempting to come up with a proof that all finite fields of the same size are isomorphic. But I'm getting stuck at a certain point, ...
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How many elements in the finite field $F_{256}$ satisfy $x^{103}=x$?

How many elements of the finite field $\mathbb{F}_{256}$ with 256 elements satisfy $x^{103}=x$?
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2answers
160 views

Cycling through powers of a generator of finite field. Something similar to modpow for Z/nZ

I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $$g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$$ ...
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Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the field with $p$ elements. How many elements of $\mathbb{F}_p$ have cube roots in $\mathbb{F}_p$? I had this question on an exam and after ...
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Could you show me an example of an order 7 pandiagonal latin square?

Could you show me an example of an order 7 pandiagonal latin square? A pandiagonal latin square is one where no broken diagonal contains repeated symbols. I have found examples for smaller order but ...
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Galois Field GF(4)

Question: Why is the table of GF(4) look like the one below? I know it has to do with the fact that 4 is composite Let GF(4) = {0,1,B,D} Addition: $$ \begin{array}{c|cccc} + & 0& 1& ...
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Question about the degree of the inverse of a polynomial $p(x)$ modulo $q(x)$

I have a doubt about the modular inverse polynomial degree. Let $p(X)$ be a polynomial in the ring $F[X]$ with $\deg(p)=\delta$, where $F$ is a finite field with characteristic 2. If ...
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For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$?

For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$ ? E.g. $x+1$, $x^2+x+1$ are irreducibles. Subcase of this question Factor by irreducible ...