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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Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
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25 views

$+$ and $\times $ operations in finite fields are $+$ and $\times $ $mod$ some number

I don't know how to prove this: Addition and multiplication operations in finite fields addition and multiplication $mod$ some number. Also I have another doubt that as we know that a field acts ...
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18 views

Roots of irreducible polynomials in a finite field

If $f$ and $g$ are irreducible polynomials over a finite field $\mathbb F_q$, both of degree $d$, then they both split in $\mathbb F_{q^d}$. One way to represent $\mathbb F_{q^d}$ is to adjoin a root ...
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1answer
28 views

If $K$ finite field of order $p^8$ where $p\ne3$ then $\sum_{\alpha \in K}{\alpha^2} = 0$

Let $K$ be finite field of order $p^8$ where $p\ne3$ is a prime. Show that $\sum_{\alpha \in K}{\alpha^2} = 0$.
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53 views

Square root for Galois fields $GF(2^m)$

Can we define a function similar to square root for $G = GF(2^m)$ (Galois field with $2^m$ elements) as $\sqrt{x} = y$ if $y^2 = y \cdot y = x$ ? For which elements $x \in G : \exists y \in G : y^2 = ...
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1answer
38 views

Field extension of degree 3 and polynomial roots

Deleted the old question, because tho whole question kind of changed. I am facing following problem: Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ ...
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1answer
20 views

number of quadratic residues in finite field

Is there a way to determine how many quadratic residues are there in the finite field $F_q$ for $q = p^k$? It seems if $q=p$ exactly $(p-1)/2$ are residues and the same amount are not. Does analogy ...
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1answer
42 views

Irreducible polynomial with LI roots

I am doing a small project under the domain cryptography. Recently I stuck with a problem in mathematics related to finite fields. My question is how can I found out irreducible polynomial(or ...
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2answers
59 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
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1answer
31 views

Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
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2answers
58 views

Jordan form of a Matrix with Ones over a Finite Field

Question: Find the Jordan Form of $n\times n$ matrix whose elements are all one, over the field $\Bbb Z_p$. I have found out that this matrix has a characteristic polynomial $x^{(n-1)}(x-n)$ and ...
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3answers
58 views

Can there be a Finite Field That Does Use Not Modular Arithmetic?

This may be a rather silly question, but I'm puzzled that (at least so far as I can tell) all finite fields use modular arithmetic. Is there no other way to construct a finite field than by defining ...
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32 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
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20 views

How to show that the normalizer N of a Sylow $p$-subgroup P of $S_p$ in $S_p$ is the affine linear group AGL(1,$p$)?

I want to prove this as a special case of another fact that the normalizer of G in the group of set-bijections of G is isomorphic to the semi-direct product of Aut(G) and G itself. How do I link up ...
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1answer
55 views

Operations in $\mathbb{F}_{32}$

I've some difficulties about sums in the field $\mathbb{F}_{32}$. In particular I'm studying an example of a cryptographic attack, where there are a lot of sums in this field, which I don't ...
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13 views

Is a Spread Unique?

Let $V$ be a vector space of dimension $n$. It is well known that when $r | n$, there is a set of disjoint $r$-dimensional subspaces of $V$, which covers $V$, called Spread. My question is that is a ...
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2answers
85 views

Efficiently factoring polynomials over $\Bbb F_2$

I am attempting to write some software which is intended to generically answer the question of which Cyclic Redundancy Code (CRC) generating polynomial is used for a given set of sample messages using ...
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1answer
42 views

Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
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50 views

How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that: why all primitive polynomials $\;$ $f \in F[X]$ $\;$ must be an irreducible. Note: Polynomial primitive is an ...
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1answer
50 views

Binomial Congruence (mod 5) Identity

I've got a (hard?) Putnam-style problem that I've been given to look at . . . I've never worked any problem even vaguely like this, but my director thinks I should be able to do it. I doubt it (100% ...
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49 views

Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
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111 views

Show that there is no surjective ring homomorphism from $\mathbb Z_2[x]$ to $\mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2$

I saw this question as a bonus from a past exam, and here's my solution for verification. I argued like so. I said suppose there is such a surjective homomorphism $f$, then $f(0)=(0,0,0)$, $f(1)= ...
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34 views

Finding the order of elements in a Galois Field

Does there exist a Galois field GF(4)? GF(4)={0,1,2,3}; If we take this Galois field, then the element '2' is not having any degree..? So is it possible to construct GF(4) ?
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1answer
29 views

Algorithm to multiply nimbers

Let $a,b$ be nimbers. Is there an efficient algorithm to calculate $a*b$, the nim-product of $a$ and $b$? The following rule seems like it could be helpful: $$ 2^{2^m} * 2^{2^n} = \begin{cases} ...
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What do the elements of the field $\mathbb{Z}_2[x]/(x^4+x+1)$ look like? What is its order?

Background: I'm looking at old exams in abstract algebra. The factor ring described was described in one question and I'd like to understand it better. Question: Let $F = \mathbb{Z}_2[x]/(x^4+x+1)$. ...
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109 views

Splitting $\Phi_{15}$ in irreducible factors over $\mathbb{F}_7$

I have to split $\Phi_{15}$ in irreducible factors over the field $\mathbb{F}_7$. It has been a while that I did this kind of stuff, and to be the honest, I've never really understood this matter. I'd ...
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21 views

Computation of the 2-torsion group of an elliptic curve

I have some troubles solving the following problem: Let $E$ be the elliptic curve $E:y^2+2y=x^3+x+9$ over $\mathbb{F}_{16}$. Compute the 2-torsion group $E[2]$, i.e. find all the points of order $2$ ...
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5answers
75 views

The inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/( x^3+2x^2+2)$

What is the independent coefficient in the inverse of $2x^2+2$ in $\mathbb{Z}_3[x]/(x^3+2x^2+2)$ ? I have been calculating some combinations, but I don't know how I can calculate the inverse.
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Prove the number of unordered pairs of linearly independent elements

Let $V$ be a vector space over $K$. Let $K={\mathbb{Z}}/{p\mathbb{Z}}$, and $\dim V=3$. We know that $V$ has $p^3$ elements. I need to show that the number of unordered pairs of linearly ...
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Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.

We have the Trace map defined by: $$ \mathrm{Tr}\colon \mathbb{F}_q\rightarrow\mathbb{F}_q\colon x\mapsto x+x^p+x^{p^2}+\cdots+x^{p^{n-1}}, $$ where $q=p^n$. Now I have to prove that if ...
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1answer
40 views

Proving that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$

if $f(x)$ is a cubic irreducible polynomial over $\mathbb Z_3$, prove that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$ Attempt: $f(x) = \alpha ...
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Presence of non square elements in $GF(p)$

I came across a problem which had a line of explanation as follows : Let $a \in GF(p). a$ is a non square in $GF(p),~ p \neq 2 \implies \nexists~ b\in GF(p)~~|~~a =b^2 $. But, is it really possible ...
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65 views

Every irreducible polynomial of degree $m$ over $\mathbb F_p$ divides $x^{p^m}-x$

We consider $F=\mathbb F_p$ for $p$ prime, $f(x)$ an irreducible polynomial of degree $m$ over $F$ and $g(x)=x^{p^m}-x$. I want to show that $f(x)\mid g(x)$. From the fact that the field ...
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59 views

Why is $1 + 1 = 0$ in $\{0, 1\}$ (binary field) and not 1 or 2?

Stuck on the simplest case in my foray into fields... I know there is a really similar question out there, but I can't find any contradiction with the field axioms if 1 + 1 = 1 instead of 0. Can ...
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1answer
52 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
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1answer
160 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum involving $p$-th roots of unity (where $\frac{1}{t}$ is to be understood as the field inverse $t^{-1} \bmod p$ etc.) of the form $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
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What is rational point on elliptic curve over Galois field

It is clear what is a rational point on elliptic curve, when the curve is defined over real numbers. But if it is defined over Galois field, what is a rational point? If necessary, supply an example, ...
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21 views

find an example for an equation in finite field

Consider the extension field $F_{q^n}$ as a vector space over $F_q$, find two linearly independent elements $\alpha_1$ and $\alpha_2$ in $F_{q^n}$, such that $\alpha_{i}\neq1$, for $i=1,2$, and : ...
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36 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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52 views

Inclusion of Fields whose order is a prime power

Blue was correct, I need to fix my understanding of this: Finite fields have cardinality of a prime order because they have a prime subfield that has finite characteristic. I do not completely ...
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Automorphisms of splitting field of $x^p-x-a$

Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$. I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of ...
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1answer
53 views

order of $x \operatorname{mod} p(x)$ in $\mathbb Z_2$

I am writing a software that analyze the behavior of an LFSR given its feedback polynomial. At some point, I need to compute the order of $x \operatorname{mod} p(x)$ in $\mathbb Z_2$. In mathematical ...
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1answer
36 views

Every field with characterisitic $p$ contains the field $\mathbb{Z}_p$

I seem to hold a very loose grasp of the concept of fields - I've encountered this question: Show that every finite field with characteristic $p$ contains $\mathbb{Z}_p$ (i.e. $\mathbb{Z}_p = ...
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1answer
28 views

Associative properties with two numbers

I am working through a number theory text and I am given a set $S=\{A,B\}$ and it has the properties: 1) $A+A=A$ 2) $B+B=A$ 3) $A+B=B+A=B$ 4) $A(A)=A$ 5) $A(B)=B(A)= A$ 6) $B(B)=B$ I am to verify ...
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44 views

Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and ...
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62 views

GCD of high order polynomials(modulo large prime)

I want to solve the following question: Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...
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50 views

Raising to the power over finite fields ??

Are there any tricks with raising an element from a finite field to power. For example let $ a \in GF(p^n)$ and I want to compute $a^m$ for some $m \in \mathbb{Z}$. Is there a nice trick to do this ...
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1answer
51 views

a square in a finite field of odd order

GF(q) is a finite field of order q, where q is odd. Prove that $a\in GF(q), a\neq0$ has a root in $GF(q)$ iff $a^{(q-1)/2}=1$. I tried to prove it this way: Suppose a has a root in ...
3
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1answer
57 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...