# Tagged Questions

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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### splitting field of a polynomial over a finite field

I just realized that finding the splitting field of a polynomial over finite fields is not as "straightforward" as in $\mathbb{Q}$ I am struggling with the following problem: "Find the splitting ...
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### Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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### Show that $m(g)=0$ if $g$ is generator of $GF(p^n)$ and $m(x)$ is a prime polynomial for the given field.

As the title states I need to prove the above assertion. Any hint on how I go about this. Show that $m(g)=0$ if $g$ is generator of $GF(p^n)$ and $m(x)$ is a prime polynomial for the given field.
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### An Equality Involving Character Sums in Characteristic 2

For context: Let $E = \mathbb{F}_{2^n}$, and let $\alpha \in E$. Define $\chi(a) := (-1)^{Tr(a)}$, where $Tr$ is the absolute trace from $E$ to $\mathbb{F}_2$. For the purposes of this question, we ...
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### In NSA (ZFC+IST), what can we say about generators for $\mathbb{Z}/\nu\mathbb{Z}$ for unlimited $\nu$?

Recently I've been going through a short text on Nonstandard Analysis that uses the axiomatic approach of Nelson (Internal Set Theory - IST). Its study has led me to be curious about the properties of ...
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### When can $\mathbb{Z}/p\mathbb{Z}$ be recovered by $[0]$ and $[k ^n]$?

One can see that $$\mathbb{Z}/19\mathbb{Z}=\{[0],[1],\ldots,[18]\}=\{[0],[3^1],[3^2],\ldots,[3^{18}]\}.$$ If I have $\mathbb{Z}/p\mathbb{Z}$, $p>2$ prime, when is there a $k\in\{2,3,\ldots,p-1\}$ ...
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### Number of $m$-dimensional subspaces of $V$ is same as number of $(n-m)$-dimensional subspaces.

Let $V$ be an $n$-dimensional vector space over a finite field $F$. For $0\le m \le n$, the number of $m$-dimensional subspaces of $V$ is same as the number of $(n-m)$-dimensional subspaces. I ...
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### Which one is Faster: Factoring a polynomial of degree $c\cdot d$ or Factoring $c$ number of degree $d$ polynomials?

I consider polnomial $T(x)$ defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number (256-bit). I want to factorize the polynomials over the finit field. The dgree of $T(x)$ is ...
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### prove that homomorphism (rings) from a field to ring is bijective or the zero homomorphism.

$F$ is a field and $R$ is a ring.$\:\phi :F\rightarrow R$ is a ring homomorphism. I need to prove that it is bijective or it is $\phi =0$. I tried to use some how the fact that I have opposites in F, ...
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### When can $A$ an abelian group be made into a vector space over $\Bbb{F}_p$?

Let $\Bbb{F}_p$ be the finite field of integers modulo $p, p$ a prime, let $A$ be an abelian group. Precisely when can $A$ be made into a vector space over $\Bbb{F}_p$?
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### Question about subfield of order p

I have seen this fact:- The minimal subfield of a field F of characteristic p is the field of p-elements. To me when I hear a finite field my mind directly go to think about the field of ...
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### Suppose $V$ is a vector space over field $F$, $\operatorname{char}(F)\neq 2$. Show that $V=V^+ \oplus V^-$, details below

Let $T$ be a linear transformation $T: V\rightarrow V, T^2=I$. Define $$V^+ =\{v\in V \mid T(V)= +v \}, V^-=\{ v\in V \mid T(v)= -v \}.$$ My understanding of fields is still weak, does ...
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### Field of $4$ elements is not isomorphic to a subfield of a field with $8$ elements .

To prove that a field of $4$ elements is not a subfield of field of $8$ elements . How to start the proof I have no clue. Thanks for any help.
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### I would like to construct a group that I have written down on paper in GAP.

First, let $V=\text{GF}(2^{11})$ (the group under addition) and let $\sigma$ be the squaring map (Frobenius map). Since $p=23$ divides $2^{11}-1$ there exist a $p^{\text{th}}$ root of unity in $V$, ...
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### Is a field with cyclic multiplicative group necessarily finite? [duplicate]

It is a standard result that finite fields have cyclic multiplicative groups (the nonzero elements with respect to field multiplication). A recent discussion on this Question leads me to ask about ...
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### Characteristic property of field $K$

Given that $K$ is a field, $\text{char}(K)=p$ (where $p$ is prime) we need to show that for any integer $n$ the equality $$(a+b)^{p^n} =a^{p^n}+b^{p^n}$$ We have the following ...
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### Why are there 8 elements in Galois ring problem?

Here is the problem: R=GF(2)[x] mod x^3 + 1 = 0 Now, I know these are the 8 elements in it. 0, 1, x, x+1, x2, x2 + 1, x2 + x, x2 + x + 1 According to the ...
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### Confusion about the proof that the multiplicative group of a finite field is cyclic

$F$ is a finite field and $F^{*}$ is the multiplicative group of $F$. Then $F^{*}$ is cyclic. The method that they use here to prove it is that if for each $d$ such that ...
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### Explicit construction of a finite field with $p^n$ elements

I'm trying to understand the following explicit construction of a finite field with $p^n$ elements where $p$ is prime, $n \geq 2$: Take any irreducible polynomial $f(X) \in \mathbb{F}_p$ of degree ...
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### How to find a primitive polynomial for the construction of a Galois field $\mathbb{F}_{p^m}$?

Say I need to construct the finite field $\mathbb{F}_{5^3}$ or $\mathbb{F}_{7^2}$, where do I get a primitive polynomial? Sorry I'm new to learning this and some of what I'm typing may not be making ...
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### Constructing add/multi tables for GF(2)

I started learning about finite fields and came across this problem online and I haven't seen this format before: R=GF(2)[x] mod x^3 + 1 = 0 What is the x part for? The closest I've seen is GF(3) ...
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### Counting $K_4$'s in a Paley Graph

Let $p \equiv 1 \pmod{4}$ be prime, and let $G$ be a graph such that $|V(G)| = p$ and $\{u,v\} \in E(G) \Longleftrightarrow u-v \equiv x^2 \pmod{p}$ for some integer $x$. How many times does $G$ ...
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### How to compute the probability distribution of $w=r_1\cdot a+r_2$ in $\mathbb{F}^*_p$

Hypothesis: Let $\mathbb{F}^*_p$ be a finite field, where $p$ is a prime number. $r_1,r_2$ are elements of the field picked uniformly at random. Let $\alpha$ be a fixed element of the field. ...
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### How does any function of a masked value change its distribution?

My question is related to information security, in particular data integrity. Consider a client has a fixed value $y$ and two uniformly random values $a$ and $b$. It computes $v=a\cdot y+b$. Note ...
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### What does $\mathrm{M}_{2}(\mathbb{F}_{7})$ mean in terms of matrix fields

I just need to know what exactly this means
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### Degree of splitting field of polynomial over a finite field

Let $f$ be a polynomial over a finite field $F$ which decomposes into a product of irreducible factors $f=p_1...p_k$ of degree $n_1,...n_k$. How can I prove that the degree of splitting field of $f$ ...
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### How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
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### How To Prove that $v=r\cdot z\$ is Distributed Uniformly at Random [duplicate]

I consider a finite field $\mathbb{F}_q$, where $q=2p+1$, and $p$ , $q$ are prime numbers. Let $z$ be a fixed element of the field. Also let $r$ be a value picked uniformly random from the field ...
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### Is $v=\ (r_i)^{-1}\cdot z$, a uniformly random value of a field?

We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers. Let $r_i$ be a value picked uniformly at random from the field such that $r_i>\frac{q}{2}$. Let $z$ be a ...
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### Probability and Data Integrity

This question is about probability and Security (i.e. data integrity). The scenario I am going to explain is a client-server case where the server may modify the client's data. We define a field ...
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### Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$. Attempt: I am confident that I computed $|E|$ correctly, but I am ...
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### Irreducible linear set of quadratics over $\Bbb F_p$

Given $a,b\in\Bbb F_p$, denote $$S(a,b)=\big\{(a+\beta)x^2+(b-\beta)x+1\in\Bbb F_p[x]:\beta\in\Bbb F_p\big\}.$$ Denote $$S(a,b)_\mathrm{red}=\big\{g(x)\in S(a,b):g(x)\text{ is reducible}\big\}.$$ ...
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### Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$

Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$ I am trying to solve this question which looks like to be ...
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### Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
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### Finding $|E|$, where $E$ is the Splitting Field of $x^8-1$ over a Field of $4$ Elements.

This is my attempt to find $\vert E \vert$, which is the order of the field $E$. If I am on the wrong track, please guide me to a technique that will work with more general fields and polynomials. We ...
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### Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
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### When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?
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### homogeneous polynomials over finite fields

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?
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### Exercise in arithmetic of a finite field

I am in difficult in resolving this exercise in Galois Theory : "in $GF(2^5)$ calculates the product $(1,1,1,0,1)(0,1,0,1,0)$ , generator of $GF(2^5)^*$ ". I don't know how to proceed.. thank you
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### The cardinality of elliptic curves over finite field

Given an elliptic curve over $\mathbb Q$ as $y^2=f(x)$ where $f(x)$ is a cubic polynomial. In some places I read that if $p$ is a prime of good reduction then we have that $E(\mathbb F_p)=p+1$. Is ...
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### How many elements does this ring have?

I know that the following ring is not a field because the defining polynomial is reducible into two polynomials that are irreducible: $$\mathbb Z_2[X]/(x^5+x+1)$$ where x^5 + x + 1 = (x^2 + x + 1) ...
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### Number of subfields of a given field [duplicate]

Let $F$ be a field with $5^{12}$ elements. What is the total number of proper subfields of $F$? A) $3$ B) $6$ C) $8$ D) $5$ Explain the concept used to solve the question.
### Difference between $f(.)$ over $\mathbb{Z}_q$ and $f(x)$ for $x \in\mathbb{Z}_q$
What is the difference between the two following notations. function $f(.)$ over $\mathbb{Z}_q$ . function $f(x)$ where $x$ takes values from $\mathbb{Z}_q$ I think that both are same. Is there ...
I want to show that $\sum_{x\in \mathbb F_{q}}x^i=0$ if $q-1$ does not divide $i$. Can someone give me a hint?