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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Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
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Showing that $c_{i}\equiv 0\pmod{p}$

Let the numbers $c_{i}$ be defined by the power series identity $$\frac{1+x+x^{2}+\ldots+x^{p-1}}{(1-x)^{p-1}}= 1+c_{1}x+c_{2}x^{2}+\ldots$$ Show that $c_{i}\equiv 0\pmod{p}$ for all $i\geq 1$. $\...
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How to evaluate GF(256) element

I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...
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Is $Z_q[X]/(\phi(X))$ a field?

Let $\phi$ be an irreducible polynomial and $q$ a prime number. Let $R=Z_q[X]/(\phi(X))$ be the ring of polynomials modulo $\phi$ with the coefficients in $Z/qZ$. I wonder why $R$ is referred always ...
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How to make the probability that two random sets have any intersection close to zero (negligible)?

This question is related to one of my question: Probability that two random sets have at least one element in common Assume we have a field $\mathbb{F}_p$, where $p$ is a large prime number i.e. $...
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Extending a code by adding a parity check

Let $C$ be a $[n,k,d]_2$ code where $d$ is odd. It is known that you can construct a $[n+1,k,d+1]_2$ code by adding a column $\boldsymbol{c}_{n+1}$ to the codebook matrix where each element contains ...
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How to represent $GF(q)$ element in Matlab [on hold]

Everyone, I am trying to simulate in Matlab the Galois field $GF(q)$ where $q$, for example, is $8$. I know the elements of $GF(8)$ are $0,1,\alpha,\alpha^2,\alpha^3,\ldots,\alpha^6$. Also, we can ...
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Counting GF($q=8$) matrices with a certain property

Let us denote by $\boldsymbol{v}_i$ the columns of an $m \times n$ GF($8$) matrix. The field elements are enumerated $\{0,1,2,...,q-1\}$. To define the arithmetic operations between field elements, we ...
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Find roots of polynomial in a finite field

I need to build a field $L$ of 121 elements and find how many roots polynomial $g=x^9-1$ has in $L$. Then to find all these roots. So, $121=11^2$ this is power of prime. We can build finite field of ...
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What is the meaning of $\Bbb{Q}[x]/f(x)$?

I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$? If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of ...
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Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
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48 views

A question on finite fields.

Suppose you have a finite field $\mathbb{F}$, where $|\mathbb{F}|=p^n$ and $p$ is a prime number. Also suppose that $f(x)=x^2+b\in \mathbb{F}[x]$ is an irreducible polynomial over $\mathbb{F}$ and $r$ ...
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Sum of the divisors of polynomials in $\mathbb{Z}_2[x]$ [closed]

Let $A$ be a polynomial in $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors of $A$. Let $A=x^h(x+1)^kP^lQ^{2n-1}$, where $P,Q$ are irreducible polynomials with degree $\geq 2$, $l\neq 2^r-...
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Field extension $\mathbb F_p\subset E$ [closed]

Suppose there exists a field extension $\mathbb F_p\subset E$. Question: Is it possible that the degree is $[E:\mathbb F_p]=2$. And how many elemnts are in E then? How can I proof such a question?...
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Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...
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Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
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Is $\mathbb{Z}_p$ a field? (p prime)

I was wondering if $\mathbb{Z}_p$ ($p$ prime) was a field, because in some notes I read there's written that $\mathbb{F}_p = \mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$ is a "prime subfield" But I was ...
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Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

The usual answer will go like this: Since $2 \not | \ 3$ and $\mathbb{F}_{p^r} \subset \mathbb{F}_{p^s}$ if and only if $r | s$, then $\mathbb{F}_9 \not\subset \mathbb{F}_{27}$. However, I'm ...
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Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
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Number of even irreducible monic polynomials of a given degree over a finite field

It is well-known that the number of irreducible monic polynomials of degree $n$ over the finite field of $q$ elements is given by the formula $$\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^{d}...
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Irreducible polynomial on $\mathbb{Z}_2$-field

I've found a theorem in the book "Linear Groups" (Dickson, 1901, p.16.): "In $\mathbb{Z}_2$, the degrees of the irreducible divisors of $x^{2^m}-x$ are divisors of $m$." I've read the prove in this ...
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irreducibility of a bivariate polyonimal over a finite field

Let $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the bivariate polynomial $$P(x,y)=y^2- x(x-1)(x-a)(x-b),$$ where $a\neq b$, and $a,b \neq 0,1$ are some arbitrary elements of $\...
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Characterising maximal ideals in $\mathbb F_p[x]$, $\mathbb Z[x]$

I'm interested in characterising maximal ideals in $\mathbb F_p[x,y]$. More precisely, my problem is: Find all possible cardinalities for fields of the kind $A/I$, where $A=\mathbb Z[x,y]/(x^2+y^2)...
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Primes is in P, proof of hendrik Lenstra Jr. lemma

In the paper describing AKS primality test : http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf On page no. 8 Lemma 4.7 last paragraph, I cannot understand how number of ...
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Find the condition of $h$ satisfy $x^3+1$ divides $x^{2h+1}+1$ over $\mathbb{Z}_2$.

Find the condition of $h$ satisfy $x^3+1$ divides $x^{2h+1}+1$ over $\mathbb{Z}_2$ and find the quotient polynomial. (Sorry for my bad English)
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Why $\mathbb{Z}_p$ can't have proper subfields?

From the notes I'm studying from I read that " $\mathbb{Z}_p=\mathbb{F}_p$ has no proper subfield." The rationale is: "assuming $\mathbb{K}$ is a subfield of a finite field $\mathbb{Z}_p= \mathbb{F}...
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What happens if I take a quotient over a reducible polynomial?

I know that for adjoinging roots to a field, I need to find irreducible polynomials so that the ideal I am taking the quotion with will be maximal, hence the resut being a field. Imagine I am working ...
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prove that $x^{2\cdot 3^n}+x^{3^n}+1$ is not primitive (mod 2)

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, It seems that there are no primitive polynomial of the form $$x^{2k}+x^{k}+1\quad k>1$$ and I'...
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Hasse-Weil bound for even degree polynomial

Consider the equation $y^2 = f(x)$, in the finite field $\mathbb{F}_q$($q$ is a prime power) where $f(\cdot)$ is a monic polynoimal of even degree (greater than or equal to $4$) with integer ...
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Field extensions and monomorphisms

I am working through some algebra exercises and got stuck with the following problem: I am working in the finite field $\mathbb{Z}_5$. Let $f \in \mathbb{Z}_5[x]$ and $f(x) = x^2 + 2$. By simple ...
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Factorization of polynomials over $\mathbb{Z}_3$

I have been given these two polynomials $$f(t)=t^3+2t+1 \text{ & }g(t)=t^3+t^2-t+2$$ the problem says, decide if both factorization fields are isomorphic. For the second polynomial I got that $$g(...
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Irreducible polynomial of every degree over finite field

The existence of polynomials in title has been asked as a problem on MathStack many times; some answers were using existence of finite bigger fields, and some answers concern Mobius function with ...
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Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

I came across the following excercise and do not know how to go about this. Given the polynomial $x^q -x -1$ in $\mathbb{F}_{q}$. Consider $q=8$. Show this polynomial is reducible by considering an ...
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Nilpotence and conjugacy in $M(p,\mathbb F_p)$

I have to solve the following problem: Characterize matrices $X\in M(p,\mathbb F_p)$ (note that $p$ is the dimension and the characteristic of the field) such that there exists $Y$ with the ...
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Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...
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Intermediate fields of a finite field extension that is not separable

Let $\mathbb{F}_p$ be the finite field with $p$ elements, where $p$ is a prime number. Let $x$ and $y$ be transcendental and algebraically independent over $\mathbb{F}_p$. The extension $\mathbb{F}_p(...
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On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
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A tip to verify property of a finite field in Linear Algebra

Let $m$ be a prime number with the following operations in the set $\mathbb{Z}_m = \{\bar{0}, \bar{1}, \dots, \bar{m - 1}\}$: $\bar{a} + \bar{b} = \bar{c}$, where $c$ is the modulus of $a + b$ by $m$...
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Problem in solving a question related to field isomorphism. [duplicate]

How many fields are there (upto isomorphism) of order 6. I dont know how to proceed. I don't know how to proceed. Please help me. Thank you in advance.
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Proving $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in a finite field

Prove that if $F$ is a field with $p^n$ elements and $\alpha,\beta \in F$, then $$(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$$ From Newton identity, we have that $$(a + b)^n = \sum_{i = 0}...
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A combinatorial question

Let us look on a $p\times p$ board (the $(\mathbb{F}_p)^2$ plane) with a single piece on the down left corner $(0,0)$. This is a special piece that has $3$ legal moves: Moving one step up $\pmod p$ ...
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Simplification of a double summation in a polynomial ring over a finite field

I am looking out to simplify the following double summation in $\mathbb{F}_q[x_1,x_2]$, where $p$ is a prime and $q=p^k$ for some positive integer $k$ and a positive integer $r$ such that $0 \leq r \...
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Why are finite field elements polynomials

Finite fields are split up into two parts. Prime fields, arithmetic is simply mod p.A prime fields takes the form $GF(p)$, where $p$ is prime. Why for extension fields, eg, of the form $GF(p^m),m>1$...
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Show $L$ can be extended to $M$ with $M/F$ cyclic

Suppose that $F$ has characteristic $p$ and $L/F$ is a cyclic extension of degree $p$. I'm trying to show that $L$ can be extended to $M$ where $F\subset L\subset M$ with $M/F$ cyclic of degree $p^2.$ ...
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Splitting field in relation to finite fields

I'm trying to prove that the subfields of a Galois Field $GF$ of order $p^n$ are isomorphic to a Galois field of order $p^r$ where $r|n$, and that there exists a unique subfield for each such $r$. I ...
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Probability that a random polynomial has a linear factor?

What is the probability that a random degree $d$ polynomial in $\Bbb F_p[x]$ has a linear factor (root in $\Bbb F_p$) for cases $d>p$ and $d<p$? Lower bound is $\frac1p$ but definitely seems ...
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Prove the polynomial $P_a=X^5 + a$ is reducible over a field

Let $(K, +, \cdot)$ a finite field so that the polynomial $P=X^2-5$ is irreducible. Prove that: a) $1+1 \ne 0$ b) The polynomial $P_a=X^5 + a$ is reducible $\forall a \in K$ a) ...
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Show $|E_1(\mathbb{F_q})|+|E_2(\mathbb{F_q})|=2(q+1)$

...under the assumption that $E_1,E_2$ are elliptic curves over $\mathbb{F_q}$ and that there is a (surjective) isogeny $\pi:E_1\rightarrow E_2$ defined over $\mathbb{F_{q^2}}$ obeying $\pi\phi_1=-\...
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1answer
56 views

Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
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what does $\alpha$ signify in finite fields modular arithmetic

Say $\frac{\mathbb{Z}_{2}\left [ x \right ]}{x^{2}+x+1}=\left \{0,1,\alpha ,1+\alpha \right \}$ is a finite field with its elements listed. I am finding it difficult to understand what it means ...