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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How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
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Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
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29 views

Creating a matrix such that all the sub-matrices are max rank

Let $A\odot B$ denote the elementwise multiplication of matrices $A$ and $B$. Given a binary matrix $B_{m \times n}=[b_{ij}]$, $b_{ij} \in \{0,1\}$, I want to find a matrix $A=[a_{ij}]$, $a_{ij}\in ...
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What is an algebraically closed field of characteristic $p$?

I suspect that this is a very simple question, but I need to ask. My question is How do the fields of characteristic $p$ look like? If $K$ is a finite field of order $p^n$, then $K$ has ...
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1answer
51 views

Basis of finite field as vector space

If we consider GF(8) as a vector space over GF(2), what are the basis for GF(8)? and How can we define a dual space for GF(8) as a vector space?
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Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
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31 views

Question on finite fields

I was curious about this question: Let $p$ be a prime, and $d \geq 1$ and $K$ is a field of $p^d$. How many proper subfields does $K$ have? All I know if that a finite field has order $p^n$, where ...
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62 views

Can we calculate $a^{n i} \mod p$?

If we have a natural $n$ (not $0$), and a prime $p$, is it possible to calculate $$a^{n i} \mod p$$ where $i$ is the imaginary number $\sqrt{-1}$? SOME THOUGHTS Knowing that $a^{i \cdot i} = ...
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28 views

Degree of a Finite Field

Consider the finite field of characteristic $\mathbb{F}_{p}$ and the polynomial $f(x) = x^{p^{n}}$ - x. The splitting field $f(x)$ is a field $\mathbb{F}_{p^{n}}$ with $p^{n}$ elements. Given this ...
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41 views

How to give irreducible factorization in $\mathbb{Z}_5$?

I have this polynomial: $$f=x^5+2x^3+4x^2+x+4$$ How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.
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Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
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31 views

What is the definition of a non-degenerate homogeneous quadratic form over a finite field?

I read in some finite geometry notes by S. Ball and Z. Weiner the following: A conic is a set of points of $PG(2,q)$ that are zeros of a non-degenerate homogeneous quadratic form (in $3$ ...
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1answer
26 views

Every irreducible polynomial in $\mathbb{F}_p[x]$ is separable?

How can I show this? I tried proving the contrapositive statement but didn't get anywhere. I think I may have to do something involving automorphisms of the splitting field of such a polynomial, and ...
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27 views

If $F$ is a finite field then there exists an irreducible polynomial in $F[x]$ with degree $n$ for all $n\in \mathbb{N}$.

If $F$ is a finite field then there exists an irreducible polynomial in $F[x]$ with degree $n$ for all $n\in \mathbb{N}$. How can I show this? A hint was given: 'Can you think of a condition that ...
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1answer
62 views

Why doesn't SAGE understand reduced expressions mod p in a finite field extension?

Suppose I have a finite field $\mathbb{F}_{13}$ and I would like to adjoin an element, $\zeta$, with order $3$, since $\mathbb{F}_{13}$ does not contain one. So consider $\mathbb{F}_{13}(\zeta)$. Then ...
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52 views

Isomorphism between finite fields

Refering to this question suppose I have $l(x):=x^3+x+1$ and $m(x):=x^3+x^2+1$. Then prove there is an isomorphism between $\mathbb{F}_3 [x]/l(x)$ and $\mathbb{F}_3[x]/m(x)$ I can say that elements ...
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Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
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61 views

Regarding doubt of order of element in a finite field

The problem goes as : Let $p$ be an odd prime & $\mathbb F_{p} =\mathbb Z/p\mathbb Z$.Show that: $x^{2}+1$ has a root in $\mathbb F_{p}$ iff $p \equiv 1 ( mod $ $4)$ . My Solution: $\mathbb ...
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104 views

Galois field theory

$\mathrm{GF}( 29^2)$ is created by adjoining the root of the irreducible quadratic $p= x^2 + 7x +15$ to the field $\mathrm{GF}(29)$ . The cubic polynomial $q = Y^3 + (26x + 26)Y^2 + (8x+22)Y +13x+23$ ...
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124 views

Multiplying in GF(128)

I know that in GF(128) $a + b = a \oplus b$. I have an multiplication table for GF(128). In this table $7\cdot 5 = 27$. How can I create table like this?
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My notes on $\Bbb{Z}/p\Bbb{Z}$-theoretic computational complexity

(Question at the very bottom) Def 1. Let $F = \Bbb{Z}_p$ be a finite field. Then an $F^k$-machine is a machine with $k$ input / output memory slots. All computations are done in the field $F$ and ...
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32 views

Subfields of irreducible polynomial fields with known dimensions

Let's say we have an irreducible polynomial, $h(x) = x^4 + x + 1 \in \Bbb F_2[x]$, and that L is a field equal to $\Bbb F_2 [x]/(h(x))$. How would I go about finding a subfield K such that $[L : K] = ...
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62 views

Do there exist polynomials not computable in polynomial time?

Motivation: Computing a problem in $k$ memory slots Do there exist polynomials in $R = \Bbb{Z}_p[z_1, \dots, z_k]$ that can't be computed in time polynomial in $k,p$? Thanks... Good luck! Edit. I ...
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Any problem computable in $k$ memory slots can be computed with polynomials.

Let our memory slots be represented by elements of $\Bbb{Z}_p$ for a prime $p$. $k$ memory slots would be $k$ copies of the ring: $R = (\Bbb{Z}_p)^k$. Suppose that for a problem $f : X \to Y$, ...
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Roots of polynomial in $F_3[x]$

Let $\alpha$ be a root of $x^2 + x + 2 = 0$ in $F_3[x]$. I am asked to show that $x^3 + x + 1$ has roots $\alpha$, $\alpha^2$ and $\alpha^4$. I started by observing that $\alpha^2 + \alpha + 2 = 0 ...
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Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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186 views

Why must a field with a cyclic group of units be finite?

Let $F$ be a field and $F^* \subseteq F$ be its group of units. If $F^*$ is cyclic show that $F$ is finite. I'm a bit stuck. I know that I can represent $F^* = \langle u \rangle$ for some $u \in F^*$ ...
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Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
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Uniqueness of projective plane of order 5

Is there a slick way to see the uniqueness of projective plane (equivalently, an affine plane) of order $5$?
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56 views

Linear independence of finite field elements and subfields

Let $q$ be a prime power and $n=lm$ an integer with $l,m>1$. We know that the finite field $GF(q^n)$ is a $n$-th dimensional vector space over $GF(q)$, and it is also a $l$-th dimensional vector ...
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48 views

Symbolic computations in finite fields of unspecified order

The general setting is that I want to multiply some matrices (to many to do it by hands) over a finite field. The problem is that these matrices depend on certain parameters taken from the field and ...
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Extension field of F2 , expressing roots and primitive elements in that field

Let $\Phi$ be an extension field of $\Bbb{F}_2$ of extension degree s >1. Let $a(x)$ be a non-zero polynomial with the coefficients in $\Bbb{F}_2$. (a) Show that if $\beta$ is a root of the ...
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How to express each element in a field F as a power of a primitive element? [closed]

I have a field F(2^4) and it is represented as a residue ring of the polynomials over F2 modulo the polynomial β4+β3+β2+β+1. I want to express each element in this field as a power of a primitive ...
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36 views

Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
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Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
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55 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
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30 views

multiplication in finite fields irreducible polynomial

I just started doing some reading about multiplication in finite fields and i keep stumbling over one point: in the field G(2^8) how does x^8 + x^4 + x^3 + x + 1 = 0 imply that x^8 = x^4 + x^3 + x + ...
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Finite field and Automorphism

Problem 1. Let S be a finite field of characteristics 2 and the map be define as $\eta$: S$\longrightarrow$S x$\longmapsto$x$^p$ Show that $\eta$ is automorphism, i.e., S is isomorphism ...
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Finite fields formed from irreducible polynomials

Let's say I have an irreducible polynomial $f(x) \in\mathbb Z/(2)[x]$ with a degree n that is at least 2 or greater. How would I go about proving that $\mathbb Z/(2)[x]/(f(x))$ is a finite field of ...
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$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
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Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
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The group structure of elliptic curve over $\mathbb F_p$

I want to find the group of the elliptic curve $y^2=x^3-x$ over $\mathbb F_p$ for all primes $p \le 29$. But I know only 1 fact about the structure of this group: $E(\mathbb F_p)=\mathbb Z/m \mathbb Z ...
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Order Of The Intersection of Two Subfields.

Last question haha, Let $E$ and $F$ be subfields of $GF(p^n)$. If $|E| = p^r$ and $|F| = p^s$, what is the order of $E\cap F$? I read a corollary that "A finite field of order $p^n$ contains a ...
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Showing an element in a Finite Field can be written as a power.

I had a question that I'm stuck with: Show that every element in $GF(p^n)$ can be written in the form of $a^p$ for some unique $a\in GF(p^n)$. So this field is the splitting field for the polynomial ...
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Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
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Generators of $\mathbb{F}_9/\mathbb{F}_3$ that do not generate $\mathbb{F}_9^{\times}$

Find a generator of the extension $\mathbb{F}_9/\mathbb{F}_3$ that does not generate the multiplicative group $\mathbb{F}_9^{\times}$. how many such elements exist? what are their minimal ...
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55 views

Construction of a polynomial

Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variable prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, ...
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Number of monic polynomials = $q^n$?

In the situation $q=p^k$ with $p$ prime and $k \in \mathbb{N}$ I have the following question: Why is the number of monic polynomials of degree $n$ in $\mathbb{F}_q[X]$ $$q^n \ ?$$
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43 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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154 views

Calculating a strange algebraic limes

I have a problem with calculating a strange limes: Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variabel prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in ...