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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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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53 views

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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39 views

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 ...
2
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2answers
125 views

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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27 views

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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38 views

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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50 views

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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1answer
39 views

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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1answer
49 views

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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174 views

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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46 views

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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27 views

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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96 views

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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63 views

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 ...
3
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30 views

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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66 views

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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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 ...
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Choose a basis of $\mathbb{F}_q/\mathbb{Z}_p$ to do inverse quickly.

Let $\mathbb{F}_q$ be the finite field with $q$ elements ($q=p^n$, $p$ is a prime). $\mathbb{F}_q$ can be regarded as a linear space over the field $\mathbb{Z}_p$ of dimension $n$. The question is: ...
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How To Prove That The Rijndael Polynomial Is Irreducible?

I am learning about the AES algorithm which uses the finite field ${\mathbb{Z}_2[x]}\over{(p(x))}$, where $p(x)=x^8+x^4+x^3+x+1$. How do you prove that this polynomial is irreducible? I know that for ...
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1answer
36 views

Sums involving binomial coefficients in a finite field

Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source ...
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1answer
40 views

Zech Logarithms

Let $a$ be a primitive element of $\mathbb{F}_{16}$ that satisfies the equation $a^4=1+a$. The logarithm of $1+a+a^2$ in $\mathbb{F}_{16}$ with base $a$ is the integer $i$ such that $0≤i<15$ ...
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Non-separable, infinite field extensions of non-zero characteristic

I have been trying to find examples (and non-examples) of fields which are separable, finite and have characteristic equal to zero. Separable Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ because the ...
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Show there exists an $\alpha\in F_{p^n}$ such that $\alpha +\alpha^p+\alpha^{p^2}+\cdots+\alpha^{p^{n-1}}\neq 0$

This is part of a problem, 7.22c in "Ireland and Rosen" (self-study). In the prior problem it is shown that for prime $p$ and $\alpha\in F_{p^n}$, $f(x)=(x-\alpha)(x-\alpha^p)(x-\alpha^{p^{2}})\cdots(...
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Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$.

Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$. I have tried this: Suppose that $K \subseteq E \subseteq K(X)$. Let $\alpha \in K$ and ...
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Is there an algorithm to find the splitting field of a polynomial over galois finite fields?

how can I find the splitting field of polynomial $x^{13}+1$ over $GF(2)$?
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Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
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Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
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$\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$

Let $\alpha$ be a zero of $f(x)=x^3+x+1 \in \mathbb{F_2}[x]$. Show that $\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$ So we need to show that $\mathbb{...
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I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
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Finding a basis for a field

I have a polynomial f(x) = $x^3+x^2+1$ in $\mathbb{Z}_5[x]$ and it is given that F = $\mathbb{Z}_5[x]$/$<f(x)>$ = $\mathbb{Z}_5(\alpha)$ where $\alpha =x+<f(x)>$. I want to find a basis ...
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55 views

Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...
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1answer
31 views

What can we say about the dimension of $[ E : F]$ if $F finite$ and $f \in F[x]$ min. polynomial [closed]

suppose I have a field $F$ and $\alpha \notin F$. $F$ is finite so $char(F)$ is some $p \in \mathbb{N}$ when I have a minimal polynomial $f_\alpha \in F[x]$ with $deg(f_\alpha)=n$ then the dimension ...
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Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: $$f(x)=\prod_{...
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There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
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1answer
41 views

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$. I have attempted to describe the Galois group, but I've become stuck, and it's entirely possible that I've made mistakes as ...
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1answer
24 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
4
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70 views

Is the primitive element of an extension of a finite field expressible as a linear combination of adjoined elements?

Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. If $F$ is infinite, the proof of the primitive element theorem gives some $c\in F$ such that $F(\alpha,\beta)=F(...