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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Proving that a Finite Field Over Its Prime Field Is Galois

In the above, I don't understand why the author needs to use another theorem to show that $f$ is separable. Theorem 3.4.5 says that in a finite field, say $F$, the Frobenius automorphism gives ...
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Irreducibility of $f(x)=x^4-x^3+14x^2+5x+16$?

Consider the polynomial $$f(x)=x^4-x^3+14x^2+5x+16$$ and $\mathbb{F}_p$ be the field with $p$ elements, where $p$ is prime. Then Considering $f$ as a polynomial with coefficients in $\mathbb{F_3}$, ...
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proving mod over GF(2)

over GF(2), $y^4 = (y^4 + 1) + 1$, so $y^4 \equiv 1 \mod y^4 + 1.$ Prove $y^6 \equiv y^2 \mod y^4 + 1$ I don't get why $y^4 = (y^4 + 1) + 1$? Shouldn't it be $y^4 \mod y^2+y+1 = (y+1)+1?$ ...
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Show that multiplicative order mod p exists and that it divides (p-1)

Let $p$ be some odd prime. Let $r$ be the smallest natural number such that $x^r \equiv 1 \pmod{p}$ for some $x \in \mathbb{F}_p^{\times}$. Prove that such an $r$ exists, and that it divides ...
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multiplication over gf(16)

Can some one show me how to do multiplication over gf(16) step by step I found this example online, http://userpages.umbc.edu/~rcampbel/Math413Spr05/Notes/12-13_Finite_Fields.html#An_Example. An ...
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How to evaluate this function in F_p efficiently?

For the positive prime integer $p$, Let $\mathbb{F}_p=\{0,1,\cdots, p-1\}$ be the finite field of order $p$. For $x\in \mathbb{F}_p$, define $f_p(x)$ to be the maximum element in the set $\{ ...
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35 views

Rank one decomposition or elementary tensor decomposition of matrices over commutative rings

I'm facing the following problem: Let $A$ be an $m\times n$ matrix over a commutative ring $R$ (to begin with, a finite field would be sufficient too) and want to compute a decomposition in terms ...
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zeta function variety

I'm trying to understand the motivation for zeta function of a variety over a finite field, that is the connection of the standard definition $$ \exp\left( \sum_{n=1}^\infty \frac{N_n t^n}{n} \right) ...
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The 8 field construction

Im thinking about 8 elemental field. I know about using polynoms, however, is there any possibility to construct the field with 8 elements, and these elements would be 0,1,a,b,c,d,e,f where all ...
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Can we describe multiplication on $\mathbb{F}_{2^n}$ as action on subsets of $n$-element set?

The symmetric difference between two set $A$ and $B$ denoted $A \triangle B$ is defined as the set $(A - B) \cup (B - A)$ or equivalently $(A \cup B) - (A \cap B)$. Some years ago I was quite excited ...
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root of irreducible polynomial

I have just started studying finite fields and I'm confused by the language around irreducible polynomial and find the following definition confusing: "If $f$ is irreducible in $\mathbb{F}_{q}[x]$ of ...
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64 views

Skew-Hermitian matrices

I have a couple of questions regarding skew-Hermitian matrices over finite fields. A matrix $A$ over $\mathbb{F}_{q^{2}}$ is skew-Hermitian if $A + A^{*} = 0$, where $A^{*}$ is the conjugate ...
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49 views

Does make sense talk about $Z_3(i)$?

My question is: Let $f(x) = x^2 + 1 \in \mathbb{Z}_3[x]$. I have to determine the splitting field of this polynomial. OK, so is intuitive to think that $i,-i$ are roots. OK, but does it make any ...
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Finite Fields. Why does it involve prime numbers only?

I'm just getting my head around the finite fields, so called Galois Fields. Why are they based on prime numbers only? any concept I'm missing?
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How to use Number Theoretic Transforms and to prove efficiency O(n log n log log n)?

I am currently researching Schönhage-Strassen algorithm for class. I am using this article to help me. However, I am stuck on page 13, paragraph 2. I tried a simple example of $x_i = ...
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How do I show that the two sequences below are short exact sequences of vector spaces?

$0 \rightarrow \mathbb{Z_2}^k \to \mathbb{Z_2}^n \to \mathbb{Z_2}^{n-k} \to 0$ $0 \to \mathbb{Z_2} \to \mathbb{F}_{2^2} \to \mathbb{Z_2} \to 0$, where $\mathbb{F}_{2^2}$ is the Galois field of size ...
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48 views

Formal Power series of finite field

I was given a question about finite field. Let $F$ be a finite field with $q$ elements, say with characteristic $p$ and $x_k$ be the number of monic irreducible polynomials with degree $k$ in $F[x]$. ...
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37 views

Appending field polynomials to an ideal produces a variety that excludes all elements from the algebraic closure of $k$.

In the paper "Algebraic Attacks on the Courtois Toy Cipher" written by M. Albrecht, he defined field polynomials and stated a corollary as follows: Definition: Let $k$ be a field with order ...
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Any difference working with matrices over fields?

Is there any difference regarding row operations and such with matrices when they are over fields? For instance, I have the following matrix over GF(3): ...
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Is $x^6-3$ irreducible over $\mathbb{F}_7$? [duplicate]

I know that $\mathbb{F}_7=\mathbb{Z}_7$, and the all possible solutions of $x^6-1=0$ over $\mathbb{Z}_7$ are 1~6, so if we let the root of equation $x^6-3$ as $t $ then the solutions of $x^6-3=0$ is ...
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Show that $K \neq F(a)$ for any $a \in K$.

Let $k$ be a field of characteristic $p >0$, let $K=k(x,y)$ be the rational function field over $k$ in two variables, and let $F=k(x^p,y^p)$. Show that $K \neq F(a)$ for any $a \in K$. For $a \in ...
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relations of galois field to bitoperators in C

Are bit operators in C corelated somehow to Galois field. Is there any literature (or beginner tutorials) on it?
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Linearized Polynomials

Let $F:=\mathbb{F}_{q^m}$ be an extension field of the finite field $\mathbb{F}_q$. A polynomial $L(x)$ is called a linearized polynomial over $F$ if $L(x)$ is of the form $$ L(x)=\sum_{i=0}^d a_i ...
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Polynomial Multiplication in GF(256)

I would like to compute the following: 10100011 * 01100011 in GF(256) using the AES irreducible polynomial. So first we get the polynomials: ...
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I have to find a splitting field of $x^{6}-3$ over $\mathbb{F}_{7}$

I want to find a splitting field of $x^{6}-3$ over $\mathbb{F}_{7}$. I learned that Finite field containing $\mathbb{F}_{7}$ is the form of $\mathbb{F}_{7^m}$ and it is normal extension. So I've tried ...
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Give $3$ examples of a field extensions which are neither normal nor separable.

Here $1)$ $F=\Bbb F_2(x)$ and consider $K=F(x^{1/6})$. Now $K/F$ is neither normal nor separable. $2)$ Let $k$ be field of characteristic $2$, let $F=k(x,y)$, let $S=F(u)$, where $u$ is a root of ...
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42 views

Square and Multiply with Polynomials

I'm trying to use the square and multiply algorithm to compute: x^11 mod x^4+x+1 in Z2[x], ie. in the Galois Field 2^4, GF(16) I believe all that I need to do is ...
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How to find the solutions of an equation in a Finite Field of cossets.

$$K=\mathbb{Z}_{3}/<f>|f(x)=1+2x^2+x^3$$ How do I find the solution of the equation $1+2y+2y^{2}=0$ in $K$? In the solution there is $w=x+<f>$, which used for writing the solutions of the ...
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Solutions of affine polynomials in characteristic $2$

I will give an example for expressing where I stuck about affine polynomials: Let $\alpha \in \mathbb F_{2^k}^*$. Let $L_{\alpha}(x)=x^4+x^2+\alpha x$ be a linearized polynomial over $\mathbb ...
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Understanding calculations of log/antilog tables of polynomials over finite field

I'm learning about finite fields and I came across this example online: http://www.csee.umbc.edu/~lomonaco/s12/443/handouts/Log-Antilog-Calculation.pdf I'm having trouble understanding what exactly ...
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Fast polynomial division algorithm over finite field

Recently, I am working on polynomial division. Suppose coefficients of all polynomials are elements in $GF(2^q)$. I want to calculate the remainder such that $f(x) = g(x)q(x) + r(x)$ (1) I searched ...
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Pairwise Coprime Polynomials

Given two polynomials $t(x) = x - r_1$ and $p(x) = x - r_2$ s.t. $r_1 \neq r_2 $ and $r_1, r_2 \in F$ where $F$ is a (finite) field. Are the polynomials $t(x)$ and $p(x)$ pairwise coprime and why? ...
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Constructing finite fields of order $8$ and $27$ or any non-prime

I want to construct a field with $8$ elements and a field with $27$ elements for an ungraded exercise. For $\bf 8$ elements: So we can't just have $\Bbb Z/8\Bbb Z$ since this is not even an ...
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$x^3+x+b$ where $b \in \mathbb F_{2^k}^*$ where $k$ is is even

Let $f_b(x)=x^3+x+b$ where $b \in \mathbb F_{2^k}^*$ where $k$ is is odd. Let $2^k=n$. Then $(n+1)/3$ of them is irreducible, $(n-2)/2$ of them have $1$ solution, $(n-2)/6$ of them have $3$ ...
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What is the complexity of finding f-reducing polynomial?

For a finite field $\mathbb F_q$, where $q$ is a power of prime. I want to know that complexity of finding f-reducing polynomials. The following is a definition of "f-reducing polynomial" Def) If $h ...
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Addition and Multiplicaiton using GF(2^4)

I am learning how to do arithmetic with Galois Fields (GF(2^4)). In particular, I am focusing on addition and multiplication. The addition problem we are looking at is: ...
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Coset representative of additive finite field over $F_p$.

Let $F_p$ be the finite field with $p$ elements, where $p$ is prime. Let $F_{p^n}$ be the finite filed with $n$ elements. I would like to know the coset representatives of the quotient group ...
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Rank of a matrix of binomial coefficients

This question arose as a side computation on error correcting codes. Let $k$, $r$ be positive integers such that $2k-1 \leqslant r$ and let $p$ a prime number such that $r < p$. I would like to ...
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Prove that $\Bbb F_2(\sqrt[3] x)$ is separable over $\Bbb F_2(x)$.

Prove that $\Bbb F_2(\sqrt[3] x)$ is separable over $\Bbb F_2(x)$. Now I was trying taking $X^3 -x\in \Bbb F_2( x)[X] $ then if I show this polynomial is separable then I am done. Now $X^3 ...
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Probability that a number is a root to a polynomial in a polynomial field

Consider a polynomial in a field $F$ which is a finite field on a prime $p$. Consider a polynomial $$a_0+a_1x+a_2x^2+a_3x^3....a_{n-1}x^{n-1}$$ The polynomial has a special property that all ...
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Fast computation to check if $x^{2^q}-x$ mod $p(x)$ is 0

Is there any fast way to check if the following equation holds? $x^{2^q}-x$ mod $p(x)=0$ Polynomials are over finite field $GF(2^q)$ I am aware of the algorithm which uses repeated squaring. This ...
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Prove $[L:\mathbb{F_p}]=n$

Let $f(x) \in \mathbb{F_p}[x] $ be an irreducible polynomial of degree $n$. Let $L$ be the splitting field of $f$. Prove $[L:\mathbb{F_p}]=n$. If $a_1,...,a_n$ are the roots of $f(x)$, then ...
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number of roots of a polynomial in finite field

Is there any way to determine the number of roots of a polynomial in finite field, more specifically, $GF(2^q)$, without actually solving the equation and find all roots?
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How do I know the number of primitive elements, which are them and the degree of extensions of Galois Groups

I am studying Galois Group and, I found some difficulties with exercises. I would like some explanation about how to understand: 1) How do I find the primitives elements of a Galois Group? Which ...
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How to check whether an ideal is prime or maximal?

The problem is to confirm that the ideal generated by $x^3+x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$ is prime and the ideal generated by $x^3-x-1$ is maximal in $\mathbb{Z}/3\mathbb{Z}[x]$. I tried to ...
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Why $Gal(GF(p^n)/GF(p))\equiv U(p^{n-1})$ is not true $?$

It is already proved that $1) \ \ $ $$Gal(GF(p^n)/GF(p))\equiv \mathbb Z_n$$ and $GF(p^n)$ is the splitting field of $x^{p^n}-x=x(x^{p^{n-1}}-1)$ over $GF(p)$. Now the second one ...
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splitting field of a polynomial over a finite field

I just realized that finding the splitting field of a polynomial over finite fields is not as "straightforward" as in $\mathbb{Q}$ I am struggling with the following problem: "Find the splitting ...
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Let $K$ be a field. Prove that $K(X,Y)$ is a finite extension of $K(X^2,Y^2)$ and find its degree.

I conjectured that $K(X,Y)=K(X^2,Y^2)[X,Y]$ but to prove that I need to show $K(X,Y)\subset K(X^2,Y^2)[X,Y]$. I'm having problems showing any element in $K(X,Y)$ is of the form $X^mY^nF$ where $F\in ...
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285 views

Does the multiplicative inverse of 3 and 6 exist in Modulo 9

As I understand a finite field of order $q$ exists if and only if the order $q$ is a prime power $ p^k $ (where $p$ is a prime number and $k$ is a positive integer). So when taking $ p=3 $ and $ k=2 ...
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Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...