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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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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Factor Ring question and finding maximal ideals of $\mathbb{Z}\times\mathbb{Z}$

What is the maximal ideal of $\mathbb{Z}\times\mathbb{Z}$? I think since $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, it seems like that ...
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Inverse of multivariated polynomials over finite fields [on hold]

I was thinking about if there is a general method, or at least case-by-case method, of expressing inverse functional compositions on Galois Fields. For instance, in $GF(2)$ how to express the inverses ...
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Order of a polynomial in $\mathbb F_q[x]$

I came across the term "order" in the context of $\mathbb F_q[x]$, specifically of irreducible polynomials. Does this mean order in the group theoretical sense? I tried to prove that every polynomial ...
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47 views

Proving the number of solutions in a finite field

Let F be a finite field containing $q$ elements. Let $a \in F$ be nonzero. If n divides $q -1$ prove that $x^n - a = 0$ has either n solutions in F or no solutions in F. I am unsure of how to begin ...
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Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
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Finite field, how to satisfy equation?

Suppose we have $\mathbb{F}_{2^5}$ defined by polynomial $x^5+x^2+1$, and (this is homework exercise, which I kinda solved) it is required to find suitable elements $b$, so that it satisfies equation ...
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How to show mutually orthogonal latin squares

I have a question concerning mutually orthogonal latin squares (MOLS). Let $ \mathbb F $ be a field of $n\in\mathbb N$ elements. For all $q\in\mathbb F \backslash \{0\}$, define $n\times n $ tables ...
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Solution of system of equations in prime fields

In 'Algebra', Artin writes that the system of equation: $$8x+3y = 3$$ $$2x+6y = -1$$ have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates ...
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$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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How to prove this is a field?

Let $F=(\mathbb{Z}/5\mathbb{Z})[x]/(x^2+2x+3)$. How do I prove $F$ is a field? I've shown its a commutative ring with an identity $\bar1$. Then we let ...
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How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials in $\mathbb Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in ...
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Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
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On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
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Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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T is multliplication by a primitive element

Consider $\Bbb F_{p^n}$ as a vector space over $\Bbb F_{p}$. Now let $T$ be a non-zero $\Bbb F_{p}$-linear map from $\Bbb F_{p^n}$ to $\Bbb F_{p^n}$. Now if $0$ and $\Bbb F_{p^n}$ are the only ...
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37 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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62 views

the Galois closure of $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$

I want to show that $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is NOT a Galois extension. Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $g(T) = ...
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about hyperplanes in finite fields

I was reading an article that said: if I have a finite field, say $\mathbb{F}_q^k$ where $q=p^n$ and $p$ is a prime; and a (k-2)-dimensional subspace, say $U\subset \mathbb{F}_q^k$ given by the span ...
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50 views

Rijndael S-Box algorithm: Can someone please explain how this code calculates the multiplicative inverse?

Wikipedia's explanation of the Rijndael Cipher's S-Box gives c code for calculating the S-Box. I've been able to calculate the S-Box values using exponent and log look-up tables to calculate the ...
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28 views

Isomorphisms between finite fields of same characteristic [duplicate]

Here I am taking the definition of isomorphism to be an injective homomorphism. Suppose we have two finite fields of the same characteristic, $\mathbb{F}_{p^n}$ and $\mathbb{F}_{p^m}$ with $m<n$. ...
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Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
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“Consecutive” square residues in odd-order finite fields

Let $\mathbb F = \mathrm{GF}(p^r)$ be finite field for $p$ an odd prime, and define $$ \begin{align} Q &= \bigl\{ u^2 \,\big|\, u \in \mathbb F^\times \bigr\} \\ N &= \mathbb F^\times ...
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$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I ...
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Differentiate polynomials in $\mathbb{Z}_2[x]$

It seems suggested that the differential of a polynomial in $\mathbb{Z}_2$ is as I would expect: $$\begin{align} &f = x^6 + x^3 + x + 1 \\ &f' = 6x^5 + 3x^2 +1 \mod 2 \\ &f'= x^2 + 1 \\ ...
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Subset of numbers analogous to primitive polynomials over finite fields

It is well known that many problems in number theory have an analogue on the ring of polynomials over finite fields and vice versa, the primes in $\mathbb{F}_q[x]$ being the irreducible polynomials. ...
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Prove that $F[x]/(f(x))$ has $q^n$ elements [duplicate]

let $F$ be a finite field of order $q$ and let $f(x)$ be a polynomial in $F[x]$ of degree $n \geq 1$. Prove that $F[x]/(f(x))$ has $q^n$ elements. I know that if a field is finite then the order is ...
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About Cayley graphs on finite fields.

If one is given $n$ vectors of length $n$ $\in \mathbb{F}_{p^k}^n$ for some prime number $p$ and $k \in \mathbb{Z}^+$ then how can one check if they are linearly independent? (the issue is if there ...
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Extensions of fields and dimension

The following is a homework question: Let M: N be a field extension, with a ∈ M algebraic over N. Show every element of N(a) is algebraic over N. Can anyone give me a strategy to approach this?
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Finite Fields (Factoring)

I am trying to factor the polynomial $x^3 + x^2 - x - 1$ in the quotient ring $\dfrac{\mathbb{F}_3[x]}{x^6 - 1}$ but I am having troubles finding a reduction. Something I have is $(x-1)^3 + x^2 - x = ...
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hyperplanes in finite vector spaces

how is an hyperplane in a finite vector space? I know a hyperplane is the kernel of a linear map, and the dimension of the hyperplane is n-1 if dimension of the vector space is n. So if I have, for ...
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Find a ring homomorphism $\tau: \mathbb{F} \rightarrow \mathbb{F}$

Just working on some exam prep questions, and I'm a bit stuck on this one. Let $ \mathbb{F} = \{ a + bX + cX^2 | a,b,c \in \mathbb{F}_2 = \{0,1\} \} $ be a ring with the operations: Addition, ...
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Algebra: Field Theory Question

I am being asked to find the following: Let $F$ denote the field $\dfrac{\mathbb{F}_2[\alpha]}{(\alpha^3 + \alpha + 1)}$. Simplify $\alpha(\alpha + 1)(\alpha + 1)$ in $F$ and calculate ...
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Generating elements of large group of units of a field.

Suppose $F$ is a large but finite field, and $F^\text{x}$ is the group of non-zero elements of the field. It is known that this group is always cyclic, and we could ask a natural question "what are ...
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Orthogonal matrices over GF(2^t) with the first column fixed

I'm interested in the existence of certain orthogonal transformations over $\mathbb F = \mathrm{GF}(2^t)$: matrices $M: \mathbb F^d \to \mathbb F^d$ for which $M^{\!\!\;\mathsf T\!\!\;} M = I_d$ over ...
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What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?

Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...
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Greatest common divisor of polynomial in Finite Field(256), AES

Have assigment and will use it as example, found solution computationaly, want to understand idea. It is about SubBytes procedure in AES, particulary about finding inverse of polynomial. Suppose we ...
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Equivalence of $\mathbb{F}_p$ and $\mathbb{Z}/p\mathbb{Z}$

I feel like I have some fundamental misunderstanding and I'm not really sure how to phrase this question, but here's a first attempt. In Child's Concrete Introduction to Higher Algebra (ISBN: ...
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Suppose that $a$ and $b$ belong to a field of order $8$ and that $a^2 + ab + b^2 =0$ then $a=0$ and $b=0$ . [duplicate]

Suppose that $a$ and $b$ belong to a field of order $8$ and $a^2 + ab + b^2 =0$. Then $a=0$ and $b=0$. Do the same when the field has order $2^n$ with $n$ odd? If one of the term is zero, i.e. ...
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I have to show this polynomial is irreducible. [closed]

Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
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Determine with proof the no of monic irreducible polynomials of prime degree $p$ over $F$,

Let $F$ be a field with $|F|=q$. Determine with proof the no of monic irreducible polynomials of prime degree $p$ over $F$, where $p$ need not be the characteristic of $F$. I know that $x^{p^n} -x$ ...
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Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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A question about the additive and multiplicative group of a finite field

Suppose $(F, +,.)$ is the finite field with $9$ elements. Let $G = (F,+)$ and $H=(F-\{0\},.)$ denote the underlying additive and multiplicative groups respectively. Then $G \cong \mathbb Z_3 \times ...
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Shortened Reed-Solomon proving p(D) is primitive

Assume we have a shortened $(n=18, k=12, t=3)$ Reed Solomon code in $GF(2^{8})$.Let $\alpha$ be a primitive element of $GF(2^{8})$. Consider the primitive polynomial given by: $p(D) = D^{8} + D^{4} + ...
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Why are the elements of GF(q), whose characteristic is 2, all squares? [duplicate]

If $a\in GF(2^n)$, then there is the element in $GF(2^n)$ such that $x^2 = a$. Why?
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Is there any definition of such semi-bilinear?

$K$ is a finite field which not equal to its base field $F_2$. Let $f: V \rightarrow K$ be a function and $B(x,y)=f(x+y)+f(x)+f(y)$ such that $B(x+y,z)=B(x,z)+B(y,z)$ and $B(z,x+y)=B(z,x)+B(z,y)$ for ...
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Trace over $\mathbf{F}_2$ of $1/(\alpha + \alpha^{-1})$ where $\alpha^{2^n+1} = 1$

Let $n$ be an integer $\ge 2$. Put $K = \mathbf{F}_{2^n}$ and $L = \mathbf{F}_{2^{2n}}$. Let $\alpha$ be an element of $L$ such that $\alpha^{2^n+1} = 1$ and $\alpha \ne 1$. Using Sage, I have noticed ...
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Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
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Counting total number of monic irreducible polynomials of all degrees $k$ that divide $m$.

Why is the following relation counting monic irreducible polynomials of all degrees $d$ that divide $m$ true? \begin{equation} \sum_{d\ |\ m}\left(\frac{1}{d} \sum_{c\ |\ d} \mu(d/c)\ p^{c}\right) = ...